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Page 1: Basic Definitions of Geometry - Ms. Talhami...1 Basic Definitions of Geometry Labeling: Now, let's take a look at how figures are labeled in geometry, and the meanings of these labels

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Page 2: Basic Definitions of Geometry - Ms. Talhami...1 Basic Definitions of Geometry Labeling: Now, let's take a look at how figures are labeled in geometry, and the meanings of these labels

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Basic Definitions of Geometry

Labeling: Now, let's take a look at how figures are labeled in geometry, and the meanings of these labels. This information should sound familiar. Lines:

Lines are traditionally labeled by expressing two points through which the line passes.

Lines may also be labeled with a single scripted letter, and referred to by that name.

Closed Figures:

When drawing rectangle ABCD: the letters must follow, in order, around the outside of the figure. You may start at any vertex point.

When drawing rectangle ABCD: you may label in either a clockwise or counterclockwise direction around the outside of the figure.

When drawing rectangle ABCD: you may NOT label "across" the figure as shown here. This is not rectangle ABCD. (It is rectangle ACBD.)

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Angles:

∠ABC or ∠CBA

Angles are labeled by specifying 3 points, with the center point being the vertex of the angle. This angle is NOT ∠BAC.

∠A

Angles may be labeled with a single letter at the vertex, as long as it is perfectly clear that there is only one angle at this vertex.

∠A or ∠

Angles may be represented by a single lower case letter or by a Greek letter, as long as it is clear which angle is being referenced.

∠1 and ∠2

Angles may also be represented by numbers, as long as it is clear to which angle the number applies.

Congruent angles are angles that have the same measure.

∠ABC ≅ ∠DEF the angles are congruent

m∠ABC = m∠DEF the measures of the angles are equal Bisector of an angle is a ray whose endpoint is the vertex of the angles and that divides that angle into two congruent angles. Classifying Angles According to Their Measures

§ An Acute angle is an angle whose degree measure is greater than 0 and less than 90o § A Right angle is an angle whose degree measure is 90o § An Obtuse angle is an angle whose degree measure is greater than 90 and less than 180o. § A Straight angle measures 180o.

Triangles:

or any other three letter combination of A,B and C will apply to this triangle.

When using letters to refer to the sides of a triangle, it is customary to label the sides as small case letters. Across from the vertex labeled capital A will be the side labeled small case a, and so on.

A right triangle is designated with a "box" drawn in the location of the right angle.

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Never assume that diagrams are drawn “to scale” Notations(The figure vs its measures)

There are notations that refer to the actual geometric figures, and there are notations that refer to the measures (sizes, lengths) of the figures.

The bar over the letters means you are referring to the segment itself (the actual physical segment).

The letters, without a bar on top, refer to the length of the segment from point A to point B.

The angle symbols,

, mean you are referring to the angle itself (the actual physical angle).

The m in front of the angle notation refers to the measure of the angle labeled A, B and C (with vertex at B).

By definition, the term congruent means "having equal length or measure".

Segments are congruent. Angles are congruent. Triangles are congruent. The congruent symbol is used when referring to the actual physical entities (diagrams).

When referring to a length or measure, the equal sign should be used. You speak of numbers as being equal (or not equal).

(Triangles are not referred to as being "equal", since they are not numbers, and we do not speak of the measure of a Δ.)

Undefined Terms: There are three words in geometry that are not formally defined. These words are point, line and plane. While these words are "undefined" in the formal sense, we can still "describe" these words.

POINT • a point indicates a location (or position) in space. • a point has no dimension (actual size). • a point has no length, no width, and no height (thickness). • a point is usually named with a capital letter. • in the coordinate plane, a point is named by an ordered pair, (x,y). While we represent a point with a dot, the dot can be very tiny or very large. Remember, a point has no size.

LINE (straight line) • a line has no thickness. • a line's length extends in one dimension. • a line goes on forever in both directions. • a line has infinite length, zero width, and zero height. • a line is assumed to be straight. • a line is drawn with arrowheads on both ends. • a line is named by a single lowercase script letter, or by any two (or more) points which lie on the line.

PLANE • a plane has two dimensions. • a plane forms a flat surface extending indefinitely in all directions. • a plane has infinite length, infinite width and zero height (thickness). • a plane is drawn as a four-sided figure resembling a tabletop or a parallelogram. • a plane is named by a single letter (plane m) or by three coplanar, but non-collinear,* points (plane ABC).

Collinear points are points that lie on the same straight line. Coplanar points are points that line in the same plane.

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Complementary Angles are two angles whose sum is 90 degrees. Supplementary Angles are two angles whose sum is 180 degrees. A Linear Pair are two adjacent angles whose sum is 180 degrees. Vertical Angles are two angles in which the two nonadjacent angles (opposite) formed when two lines intersect. They are always congruent.

Vertical angles are located across from one another in the corners of the "X" formed by two straight lines. In the diagram at the right, lines m and n are straight: ∠1 and ∠2 are vertical angles. ∠3 and ∠4 are vertical angles.

1 – 3 Solve the following 1. x = ______ y = ______

2. x = ______ y = ______

3. x = ______ y = ______

4 – 8 Are the following statements true or false? 4. ∠5 and ∠3 are vertical angles T or F 5. ∠1 and ∠5 are a linear pair T or F 6. ∠4 and ∠3 are adjacent angles T or F 7. ∠4 and ∠1 are vertical angles T or F 8. ∠3 and ∠4 are a linear pair T or F 9. If ∠A and ∠B are supplementary and m∠A = 150o, what is m∠B?

10. If ∠A and ∠B are complements and m∠A = 27o, what is m∠B?

11. If ∠A and ∠B are vertical angles and m∠A = 36o, what is m∠B?

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12. If ∠A and ∠B are a linear pair and m∠A = 2x + 8 and m∠B = 3x + 2, what is the value of x?

13. If ∠A and ∠B are vertical angles and m∠A = 7x - 5 and m∠B = 4x + 10, what is the value of x?

14 – 23 Circle True or False 14. ∠1 = ∠4 T or F 15. ∠6 = ∠16 T or F 16. ∠3 = ∠5 T or F 17. ∠4 = ∠5 T or F 18. ∠2 = ∠10 T or F 19. ∠9 = ∠15 T or F 20. ∠12 = ∠14 T or F 21. ∠9 = ∠11 T or F 22. m∠11 + ∠15 = 180o T or F 23. m∠1 + ∠8 = 180o T or F

Angles formed by Parallel Lines Cut by a Transversal

m II n and l is the transversal

Congruent Angles • Vertical Angles ∠1 and ∠4, ∠2 and ∠3, ∠5 and ∠8, ∠6 and ∠7 • Corresponding Angles ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8 • Alternating Interior Angles ∠3 and ∠6, ∠4 and ∠5 • Alternation Exterior Angles ∠1 and ∠8, ∠2 and ∠7

Supplementary Angles

• ∠3 and ∠5, ∠4 and ∠6, ∠1 and ∠7, ∠2 and ∠8

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1 – 6 Solve for the unknown values: 1. x = ________

2. x = ________

3. x = ________

4. x = ________

5. x = ________

6. x = ________

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7 – 12 Solve the following: 7. m∠1 = _______ 8. m∠2 = _______ 9. m∠3 = _______ 10. m∠4 = _______ 11. m∠5 = _______ 12. m∠6 = _______ 13. Solve the following: x = _________ y = ________

14. Solve the following: x = _________ y = ________

Find the value of x and name the type of pairs of angles they are: 1.

2.

3.

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4.

5.

6.

7.

8.

9.

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9

10.

11.

12.

13.

14.

15.

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16.

17.

18.

19.

20.

21.

j

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A Triangle is a polygon that has exactly three sides. Classifying Triangles According to Sides

§ No Sides equal - Scalene § 2 sides equal - Isosceles § 3 sides equal - Equilateral.

Classifying Triangles According to Angles

§ Has a right angle - Right § Has an obtuse angle - Obtuse § All acute angles - Acute § All equal angles - Equiangular

The sum of the measures of the interior angles of a triangle equals 180º. Which of the following can represent the degree measures of the three angles of a triangle? 1. 20, 100, 60

2. 55, 45, 90

3. 30, 105, 40

4. 35, 125, 10

Find the degree measure of the third angle of a triangle, if the 1st two angles measure: 5. 60, 40

6. 130, 20

7. 45, 55

8. 102, 34

9. If the degree measures of the angles of a triangle are represented by 2x, x + 10,and 3x – 10, find the value of x and the measures of the three angles.

10. If the measures of the angles of a triangle are represented by 2x + 8, x - 8 and 2x – 20. Find the measures of the three angles.

1. – 3. Determine the measure of the angle: 1. m∠C = _______

2. m∠C = _______

3. x = _________

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4 – 6 Determine the measure of the angle: 4. x = ______m∠A = _______

5. x = ______m∠B = _______

6. x = _________

7 – 10 Find the value of x: 7. m∠1 = ______m∠2 = ______

8. x = _________

9. x = _________

10. x = _________

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11. If the measures of the angles of a triangle are represented by 2x, x + 12 and 3x – 12. Find the measures of the three angles.

12. If the measures of the angles of a triangle are represented by 3x, 5x - 20 and 4x + 10. Find the measures of the three angles.

13. If the measures of the angles of a triangle are represented by 2x - 8, x + 3 and 2x – 5. Find the measures of the three angles.

14. If the measures of the angles of a triangle are represented by x - 5, 2x + 3 and 4x. Find the measures of the three angles.

15. In ΔABC, the measure of ∠A is one-half the measure of ∠B and the measure of ∠C is three times the measure of ∠B. Find the measures of each angle.

16. In ΔPQR, the measure of ∠P is twice the measure of angle ∠Q. The measure of ∠R is three times the measure ∠P. Find the measures of each angle.

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Find the degree measure of each angle of a triangle if the ratio of the measures of the three angles are as follows. Then list them in order from least to greatest. 17. 1 : 2 : 3

18. 1 : 3 : 5 19. 1 : 4 : 7 20. 1 : 2 : 6 21. 1 : 4 : 5

22. 4 : 3 : 2

23. 2 : 5 : 8 24. 2 : 5 : 2 25. 3 : 4 : 5 26. 3 : 5 : 7

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An Isosceles triangle is a triangle with two congruent sides: An isosceles triangle is generally drawn so it is sitting on its base. This may not, however, be the case in all drawings.

If two sides of a triangle are congruent, the angles opposite them are congruent. OR: The base angles of an isosceles triangle are congruent. Converse: If two angles of a triangle are congruent, the sides opposite them are congruent. Find the degree measure of the vertex angle of an isosceles triangle if each base angle measures: 1. 70

2. 35

3. 54

4. 37

5. 72.5

Find the degree measure of each base angle of an isosceles triangle, if the vertex angle measures: 6. 50

7. 70

8. 82

9. 100

10. 75

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11. m∠A = ______m∠C = ______

12. x = ______m∠C = ______

13 x =

14. x = ______

15. x = ______

16. x = ______

17. m∠1 = ______m∠2 = ______ m∠3 = ______

18. m∠1 = ______m∠2 = ______ m∠3 = ______

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19. Each of the congruent angles of an isosceles triangle measures 9 less than four times the vertex angle. Find the angles of the triangle.

20. Each base angle of an isosceles triangle has a measure that is 20 more than 3 times the measure of the vertex angle. Find the measure of the vertex angle.

21. In ΔABC, AB ≅ BC. If AB = 5x + 10 and BC = 3x + 40 and AC = 2x + 30, find the lengths of each side of the triangle

22. In ΔABC, AB ≅ BC. If m∠A = 7x, m∠C = 2x + 50, find the measures of the angles of the triangle.

23. In ΔABC, AB ≅ BC. If AB = 5x and BC = 2x + 15 and AC = 2x + 30, find AB and BC.

24. In ΔEFG, EF ≅ FG. If m∠E = 4x + 50, m∠F = 2x + 60 and m∠G = 14x + 30, find the measures of the angles of the triangle.

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An exterior angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side of the triangle.

FACTS: • Every triangle has 6 exterior angles, two at each vertex. • Angles 1 through 6 are exterior angles. • Notice that the "outside" angles that are "vertical" to the angles inside the triangle are NOT called exterior angles of a triangle.

The exterior angle of a triangle is equal to the sum of the non-adjacent (opposite) interior angles. 1. An exterior angle is formed between a side and the extension of a side. It will always be a linear pair with an internal angle. In the diagram below, ∠4 is the exterior angle. The exterior angle theorem states that the EXTERNAL ANGLE IS EQUAL TO THE SUM OF THE TWO REMOTE ANGLES. The remote angles are those interior angles that are not adjacent to the exterior angle so in this case ∠1 & ∠2 are the remote angles. m∠1 + m∠2 = m∠4, Explain why this would be true.

2 – 4 Determine the missing information: 2. m∠1 = ______m∠3 = ______

3. m∠1 = ______m∠3 = ______

4. m∠1 = ______m∠3 = ______

43

1

2

A

B

C

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Find x: 5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

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17.

18.

19.

20. Find the measure of either of the exterior angles formed by extending the base of an isosceles triangle, if the vertex angle of the triangle is 20.

21. Find the measure of either of the exterior angles formed by extending the base of an isosceles triangle, if the vertex angle of the triangle is 82.

22. Find the measure of either of the exterior angles formed by extending the base of an isosceles triangle, if the vertex angle of the triangle is 40.

23. Find the measure of either of the exterior angles formed by extending the base of an isosceles triangle, if the vertex angle of the triangle is 135.

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24. An exterior angle at the base of an isosceles triangle measures 140, find the number of degrees in the vertex angle.

25. An exterior angle at the base of an isosceles triangle measures 125, find the number of degrees in the vertex angle.

26. An exterior angle at the base of an isosceles triangle measures 135, find the number of degrees in the vertex angle.

27. An exterior angle at the base of an isosceles triangle measures 100, find the number of degrees in the vertex angle.

28. In ΔPQR, the measure of ∠P is twice the measure of ∠Q. If an exterior angle at vertex ∠R has a degree measure of 120, find m∠Q

29. In ΔABC, m∠B is four times as large as m∠A. An exterior angle at ∠C measures 125. Find the degrees m∠A.

30. An exterior angle at the base of an isosceles triangle is always 1) right 2) acute 3) obtuse 4) equal to base angle

31. In ΔTRS, ∠S is a right angle. The exterior angle at vertex ∠S is 1) right 2) acute 3) obtuse 4) straight

32. For ΔABC, m∠A = 40 and m∠B =60. The degree measure of the exterior angle at vertex C is 1) 40 2) 60 3) 80 4) 100

33. In ΔABC, an exterior angle at C measures 100 degrees and angle B measures 20 degrees. ΔABC must be 1) isosceles 2) right 3) obtuse 4) equiangular

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34. Side AC of ΔABC is extended through C to D. ∠BCD measures 108, and the measure of ∠A is twice the measure of ∠B. ΔABC must be 1) isosceles 2) right 3) obtuse 4) scalene

35. In ΔDEF, m∠D = 2x + 4, m∠E = 6x – 58. The degree measure of an exterior angle at F is represented by 5x. a) find x______ b) Show ΔDEF is a right Δ

36. In ΔABC, AC is extended through C to D. If m∠BCD = 4x – 80, ∠BAC = x – 6 and ∠ABC = x – 4, what is the value of x?

37. In ΔABC, AC is extended through C to D. If m∠BCD = 5x – 4, ∠BAC = x + 30 and ∠ABC = x + 20, what is the value of x?

38. The degree measure of the vertex angle of an isosceles triangle is 120. Find the measure of a base angle of the triangle.

39. In ΔABC, ∠A = ∠C. If AB = 8x + 4 and CB = 3x + 24, find x.

40. In an isosceles triangle, if the measure of the vertex angle is three times the measure of the base angle, find the degree measure of the base angle.

41. In a triangle, the degree measure of the three angles are represented by x, x + 25 and x – 5. Find the angle measures.

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42. In ΔPQR, m∠P = 35 and m∠Q = 85. What is the degree measure of an exterior angle at vertex R?

43. An exterior angle at the base of an isosceles triangle measure 130. Find the measure of the vertex angle.

44. In ΔABC, If AB = AC and m∠A = 70, find m∠A.

45. In ΔDEF, If DE = DF and m∠E = 70, find m∠D.

46. In ΔPQR, PQ is extended through Q to point T, forming exterior angle RQT. If m∠RQT = 70 and m∠R = 10, find m∠P

47. The measure of angles of a triangle are in a ratio of 2 : 3 : 5. Find the degree measure of the smallest angle of the triangle.

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Triangle Inequalities The sum of the length of two sides of a triangle is greater than the length of the third side.

Since 9 is the longest side of the triangle, ∠C (across from it) is the largest angle and since 5 is the shortest side of the triangle, ∠A (across from it) is the smallest angle

Since 88º is the largest angle of the triangle, RS (across from it) is the longest side and since 40º is the smallest angle of the triangle, RA (across from it) is the shortest side.

Tell whether the given lengths may be the measure of the sides of a triangle. 1) 3, 4, 5

2. 5, 8, 13

3. 6, 7, 10

4. 3, 9, 15

5. 2, 2, 3

6. 1, 1, 2

7. 3, 4, 4,

8. 5, 8, 11

9. 6, 2, 3

10. 5, 3, 7

11. 4, 6, 3

12. 9, 4, 5

13. 2, 3, 5

14. 4, 4, 8

15. 3, 4, 8

16. 5, 6, 7

17. 6, 10, 9

18. 7, 5, 8

19. 6, 13, 7

20. 2, 5, 3

1. Which set of numbers could be the lengths of the sides of a triangle? 1) 5, 5, 1 2) 3, 6, 9 3) 12, 13, 20 4) 6, 7, 13

2. Which set of numbers could be the lengths of the sides of a triangle? 1) 8, 11,19 2) 11, 5, 5 3) 3, 4, 8 4) 19, 16, 20

3. Which set of numbers could be the lengths of the sides of a triangle? 1) 4, 5, 6 2) 5, 12, 13 3) 5, 5, 10 4) 7, 8, 10

4. Which set of numbers could be the lengths of the sides of a triangle? 1) 3, 9, 14 2) 1, 2, 3 3) 3, 5, 7 4) 4, 4, 8

5. Which set of numbers could not be the lengths of the sides of a triangle? 1) 1, 1, 2 2) 2, 3, 4 3) 1, , 2 4) 3, 4, 5

6. Which set of numbers could not be the lengths of the sides of a triangle? 1) 4, 7, 9 2) 4, 8, 12 3) 9, 10, 11 4) 6, 6, 11

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7. Which set of numbers could not be the lengths of the sides of a triangle? 1) 9, 12, 19 2) 6, 8, 11 3) 7, 18, 11 4) 7, 5, 6

8. Which set of numbers could be the lengths of the sides of a isosceles triangle? 1) 15, 5, 10 2) 3, 4, 5 3) 1, 1, 3 4) 6, 6, 5

9. Two sides of an isosceles triangle have lengths 2 and 12 respectively. What is the length of the third side 1) 9 2) 8 3) 12 4) 14

10. Two sides of an isosceles triangle have lengths 4 and 8. What is the third side? 1) 4 2) 6 3) 5 4) 8

11. Two sides of an isosceles triangle have lengths 3 and 6. What is the third side? 1) 6 2) 3 3) 2 4) 5

12. Two sides of an isosceles triangle have lengths 2 and 5. What is the third side? 1) 3 2) 5 3) 8 4) 2

13. Two sides of an isosceles triangle have lengths 8 and 16. What is the third side? 1) 8 2) 14 3) 24 4) 16

14. Two sides of an isosceles triangle have lengths 7 and 15. What is the third side? 1) 23 2) 15 3) 20 4) 7

15. Two sides of a triangle have lengths 5 and 8. Which length can not be the length of the third side? 1) 5 2) 4 3) 3 4) 6

16. Two sides of a triangle have lengths 4 and 7. Which length can not be the length of the third side? 1) 11 2) 5 3) 7 4) 4

17. Two sides of a triangle have lengths 5 and 4. Which length can not be the length of the third side? 1) 3 2) 2 3) 1 4) 4

18. Two sides of a triangle have lengths 4 and 8. Which length can not be the length of the third side? 1) 6 2) 4 3) 7 4) 5

19. Two sides of a triangle have lengths 7 and 10. Which length can be the length of the third side? 1) 3 2) 1 3) 4 4) 2

20. Two sides of a triangle have lengths 1 and 4. Which length can be the length of the third side? 1) 3 2) 5 3) 4 4) 8

21. Two sides of a triangle have lengths 6 and 8. Which length can be the length of the third side? 1) 2 2) 14 3) 15 4) 7

22. Two sides of a triangle have lengths 10 and 4. Which length can be the length of the third side? 1) 6 2) 16 3) 14 4) 8

23. Two sides of a Δ are 3 and 8. The length of the third side must be greater than________ and less than_________

24. Two sides of a Δ are 3 and 7. The length of the third side must be greater than________ and less than________

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25. Two sides of a Δ are 5 and 13. The length of the third side must be greater than________ and less than________

26. Two sides of a Δ are 22 and 34. The length of the third side must be greater than________ and less than________

27. Two sides of a Δ are 10 and 14. The length of the third side must be greater than________ and less than________

28. Two sides of a Δ are 5 and 10. The length of the third side must be greater than than________ and less than________

29. Two sides of a Δ are 7 and 15. The length of the third side must be greater than than________ and less than________

30. Two sides of a Δ are 3 and 5. The length of the third side must be greater than________ and less than________

1. In ΔPQR, PQ = 5, QR = 14 and PR = 11. What is the smallest angle? 1. ∠Q 2. ∠P 3. ∠R

2. In ΔXYZ, XY = 5, YZ = 7 and XZ = 18. What is the smallest angle? 1. ∠Y 2. ∠X 3. ∠Z

3. In ΔABC, AB = 3, BC = 4 and AC = 5. What is the smallest angle? 1. ∠A 2. ∠B 3. ∠C

4. In ΔTHB, TH = , HB = and TB = 4. What is the smallest angle? 1. ∠B 2. ∠T 3. ∠H

5. In ΔKBG, KG = , KB = and TB = 5. What is the largest angle? 1. ∠G 2. ∠K 3. ∠B

6. If m∠A = 60o, m∠B = 30o and m∠C = 90o, what is the shortest side of ΔABC? 1. AC 2. BC 3. AB

7. If m∠D = 110o, m∠E = 40o and m∠F = 30o, what is the shortest side of ΔABC? 1. DE 2. EF 3. ED

8. If m∠P = 60o, m∠R = 70o and m∠S = 50o, what is the shortest side of ΔABC? 1. PS 2. RS 3. PR

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9. In ΔABC, find: a) Smallest angle______b) largest angle______

10. In ΔDEF, find: a) Smallest angle______b) largest angle______

11. In ΔPNM, find: a) Smallest angle______b) largest angle______

12. In ΔXYZ, find: a) Smallest angle______b) largest angle______

13. In ΔDNA, find: a) Smallest angle______b) largest angle______

14. In ΔSUE, find: a) Smallest angle______b) largest angle______

E

15. In ΔAEM, find: a) Smallest side______b) largest side______

16. In ΔNES, find: a) Smallest side______b) largest side______

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17. In ΔYDC, find: a) Smallest side______b) largest side______

18. In ΔZPA, find: a) Smallest side______b) largest side______

19. In ΔYES, find: a) Smallest side______b) largest side______

20. In ΔFCX, find: a) Smallest side______b) largest side______

REGENTS QUESTIONS – for triangle inequalities

1. Which set of numbers represents the lengths of the sides of a triangle? 1) {5,18,13} 2) {6,17,22} 3) {16,24,7} 4) {26,8,15}

2. Phil is cutting a triangular piece of tile. If the triangle is scalene, which set of numbers could represent the lengths of the sides? 1) {2,4,7} 2) {4,15,6} 3) {3,5,8} 4) {5,5,10}

3. Which set can not represent the lengths of the sides of a triangle?

v 1) {4,5,6} 2) {5,5,11} 3) {7,7,12} 4) {8,8,8}

4. Which set could not represent the lengths of the sides of a triangle? 1) {3,4,5} 2) {2,5,9} 3) {5,10,12} 4) {7,9,11}

5. In ΔABC, AB = 5 feet and BC = 3 feet. Which inequality represents all possible values for the length of AC, in feet? 1) 2 ≤ AC ≤ 8 2) 2 < AC < 8 3) 3 ≤ AC ≤ 7 4) 3 < AC < 7

6. If two sides of a triangle are 1 and 3, the third side may be 1) 5 2) 2 3) 3 4) 4

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7. If the lengths of two sides of a triangle are 4 and 10, what could be the length of the third side? 1) 6 2) 8 3) 14 4) 16

8. Sara is building a triangular pen for her pet rabbit. If two of the sides measure 8 feet and 15 feet, the length of the third side could be 1) 13 ft 2) 7 ft 3) 3 ft 4) 23 ft

9. The direct distance between city A and city B is 200 miles. The direct distance between city B and city C is 300 miles. Which could be the direct distance between city C and city A? 1) 50 miles 2) 350 miles 3) 550 miles 4) 650 miles

10. A box contains one 2-inch rod, one 3-inch rod, one 4-inch rod, and one 5-inch rod. What is the maximum number of different triangles that can be made using these rods as sides? 1) 1 2) 2 3) 3 4) 4

11. How many integer values of x are there so that x, 5, and 8 could be the lengths of the sides of a triangle? 1) 6 2) 9 3) 3 4) 13

12. The plot of land illustrated in the accompanying diagram has a perimeter of 34 yards. Find the length, in yards, of each side of the figure. Could these measures actually represent the measures of the sides of a triangle? Explain your answer.

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13. In the diagram below of ΔABC, D is a point on AB, AC = 7, AD = 6, and BC = 18.

The length of could be 1) 5 2) 12 3) 19 4) 25

14. A plot of land is in the shape of rhombus ABCD as shown below.

Which can not be the length of diagonal AC? 1) 24 m 2) 18 m 3) 11 m 4) 4 m

15. José wants to build a triangular pen for his pet rabbit. He has three lengths of boards already cut that measure 7 feet, 8 feet, and 16 feet. Explain why José cannot construct a pen in the shape of a triangle with sides of 7 feet, 8 feet, and 16 feet.

16. On the banks of a river, surveyors marked locations A, B, and C. The measure of

and the measure of ∠ABC = 65o.

Which expression shows the relationship between the lengths of the sides of this triangle? 1) AB < BC < AC 2) BC < AB < AC 3) BC < AC < AB 4) AC < AB < BC

17. In ΔABC, m∠A = 60o, m∠B = 80o, , and m∠C = 40o. Which inequality is true? 1) AB > BC 2) AC > BC 3) AC < BA 4) BC < BA

18. In ΔABC, m∠A = 95o, m∠B = 50o, and m∠C = 35o. Which expression correctly relates

the lengths of the sides of this triangle? 1) AB < BC < AC 2) AB < AC < BC 3) AC < BC < AB 4) BC < AC < AB

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19. In ΔRST, m∠R = 58o, m∠S = 73o and. Which inequality is true? 1) RT < TS < RS 2) RS < RT < TS 3) RT < RS < TS 4) RS < TS < RT

20. In scalene triangle ABC, m∠B = 45o and m∠C = 55o. What is the order of the sides in length, from longest to shortest? 1) , , 2) , , 3) , , 4) , ,

21. In ΔABC, ∠A ≅ ∠B and m∠C is an obtuse angle. Which statement is true? 1) and is the longest side. 2) and is the longest side. 3) and is the shortest side. 4) and is the shortest side.

22. In ΔABC, AB = 7, BC = 8 and AC = 9. Which list has the angles of ΔABC in order from smallest to largest? 1)∠A, 2) 3) 4)

23. In ΔPQR, , , and . Which statement about the angles of ΔPQR must be true? 1) 2) 3) 4)

24. In ΔABC, , , and . Determine the longest side of

ΔABC.

25. As shown in the diagram of ΔACD below, B is a point on and is drawn.

If , , and , what is the longest side of ΔABD? 1) 2) 3) 4)

26. In the diagram below of ΔABC with side extended through D, and . Which side of ΔABC is the longest side? Justify your answer.

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RegularPolygon–apolygonwhosesidesallhavethesamemeasure.For#1-8,identifyhowmanysideseachpolygonhasandstateitsname.1.

2.

3.

4.

5.

6.

7.

ExteriorAnglesofPolygons

ë Theexterioranglesofanypolygonaddupto360°.ë Aninteriorangleandanexteriorangleformalinearpair.ë Thesumoftheinterioranglesofapolygonisequaltothe

productofthemeasureoftheinteriorangle&thenumberofsides.

FindingtheExteriorAngleofaRegularPolygonForeachpolygonin#8-13,determinethesumofthe

measuresoftheexteriorangles.8.Triangle 9.Quadrilateral 10.Pentagon

11.Hexagon 12.Octagon 13.Decagon

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For#14-19,nameeachpolygonanddeterminethemeasureofoneexteriorangleofeachpolygon.14.

15.

16.

17.

18.

19.

InteriorAnglesofPolygonsë Aninteriorangleandanexteriorangleformalinearpair.ë Thesumoftheinterioranglesofapolygonisequaltothe

productofthemeasureoftheinteriorangle&thenumberofsides.FindingtheInteriorAngleofaRegularPolygon:

1) Findtheexteriorangle2) Subtracttheexterioranglefrom180

FindingtheSumoftheInteriorAnglesofaRegularPolygon:

3) Multiplytheinterioranglebythenumberofsides.For#1-6,foreachofthefollowingpolygons,findthemeasureofoneinteriorangleandthenfindthesumoftheinteriorangles.1.Triangle 2.Quadrilateral 3.Pentagon 4.Hexagon 5.Octagon 6.Decagon

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InteriorandExteriorAnglesofRegularPolygonsSummary

#Sides Name SumofExterior

AnglesOneExterior

AngleOneInterior

AngleSumofInterior

Angles

3

4

5

6

8

10

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REGENTSQUESTIONS–forpolygons1. Whichregularpolygonhasaminimum

rotationof45°tocarrythepolygonontoitself?1) octagon2) decagon3) hexagon4) pentagon

2.Aregularhexagonisrotatedinacounterclockwisedirectionaboutitscenter.Determineandstatetheminimumnumberofdegreesintherotationsuchthatthehexagonwillcoincidewithitself.

3.Thepentagoninthediagrambelowisformedbyfiverays.Whatisthedegreemeasureofanglex?1) 722) 963) 1084) 112

4.Whichrotationaboutitscenterwillcarryaregulardecagonontoitself?1) 54°2) 162°3) 198°4) 252°

5.Astopsignintheshapeofaregularoctagonisrestingonabrickwall,asshownintheaccompanyingdiagram.Whatisthemeasureofanglex?1) 45°2) 60°3) 120°4) 135°

6.OnepieceofthebirdhousethatNatalieisbuildingisshapedlikearegularpentagon,asshownintheaccompanyingdiagram.IfsideAEisextendedtopointF,whatisthemeasureofexteriorangleDEF?1) 36°2) 72°3) 108°4) 144°

7.Themeasureofaninteriorangleofaregularpolygonis108°.Whatisthenameofthepolygon?

8.Themeasureofaninteriorangleofaregularpolygonis120°.Howmanysidesdoesthepolygonhave?1) 52) 63) 34) 4

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9.Melissaiswalkingaroundtheoutsideofabuildingthatisintheshapeofaregularpolygon.Shedeterminesthatthemeasureofoneexteriorangleofthebuildingis60°.Howmanysidesdoesthebuildinghave?1) 62) 93) 34) 12

10.Aregularpolygonhasanexterioranglethatmeasures45°.Howmanysidesdoesthepolygonhave?1) 102) 83) 64) 4

11.Whatisthemeasureofeachinteriorangleinaregularoctagon?1) 108º2) 135º3) 144º4) 1080º

12.Whatisthemeasureofeachinteriorangleofaregularhexagon?1) 2) 3) 4)

13.Whatisthesum,indegrees,ofthemeasuresoftheinterioranglesofapentagon?1) 1802) 3603) 5404) 900

14.Thesumofthemeasuresoftheinterioranglesofasquareis1) 180°2) 360°3) 540°4) 1,080°

15.Thesumoftheinterioranglesofaregularpolygonis720°.Howmanysidesdoesthepolygonhave?1) 82) 63) 54) 4

16.Thesumoftheinterioranglesofaregularpolygonis540°.Determineandstatethenumberofdegreesinoneinteriorangleofthepolygon.

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REGENTS QUESTIONS – from Common Core Exams 1. InthediagramofparallelogramFREDshownbelow, isextendedtoA,and isdrawnsuchthat .If ,whatis ?1) 124°2) 112°3) 68°4) 56°

2. InparallelogramQRSTshownbelow,diagonalisdrawn,UandVarepointson and ,

respectively,and intersects atW.If, ,and ,whatis

?1) 37º2) 60º3) 72º4) 83º

3. Inthediagrambelow, , ,and .Whichstatementistrue?1) isobtuse.2) isisosceles.3) 4) isscalene.

4. StevedrewlinesegmentsABCD,EFG,BF,andCFasshowninthediagrambelow.Scalene isformed.WhichstatementwillallowStevetoprove

?1) 2) 3) 4)

5. Inthediagrambelow, intersects and atGandH,respectively,and isdrawnsuchthat

.If and ,explainwhy .

6. Inthediagrambelow,lines ,m,n,andpintersectliner.Whichstatementistrue?1) 2) 3) 4)

7. InthediagrambelowofisoscelestriangleABC, andanglebisectors , ,and aredrawnandintersectatX.If ,find .

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ADDITIONAL REGENTS QUESTIONS

3 1. The diagram below shows ΔABD, with ABC, BE ⊥ AD, and ∠EBD = ∠CBD.

If m∠ABE = 52o, what is m∠D? 1) 26 2) 38 3) 52 4) 64

2. In the diagram of ΔJEA below, m∠JEA = 90o and m∠EAJ = 48o. Line segment MS connects points M and S on the triangle, such that m∠EMS = 58o.

What is m∠JSM? 1) 163 2) 121 3) 42 4) 17

3. The angles of triangle ABC are in the ratio of 8 : 3 : 4. What is the measure of the smallest angle? 1) 12º 2) 24º 3) 36º 4) 72º

:

4. In an equilateral triangle, what is the difference between the sum of the exterior angles and the sum of the interior angles? 1) 180° 2) 120° 3) 90° 4) 60°

5. Triangle PQR has angles in the ratio of 2 : 3 : 5. Which type of triangle is ΔPQR?

1) acute 2) isosceles 3) obtuse 4) right

6. In ΔABC, m∠A = 3x + 1, m∠B = 4x - 17, and m∠C = 5x - 20. Which type of triangle is ΔABC? 1) right 2) scalene 3) isosceles 4) equilateral

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7. In ΔABC, m∠A = x, m∠B = 2x + 2, and m∠C = 3x + 4. What is the value of x?

1) 29 2) 31 3) 59 4) 61

8. In ΔDEF, m∠D = 3x + 5, m∠E = 4x - 15, and m∠F = 2x + 10. Which statement is true? 1) DF = FE 2) DE = FE 3) m∠E = m∠F 4) m∠D = m∠F

9. Juliann plans on drawing ΔABC, where the measure of m∠A can range from 50° to 60° and the measure of m∠B can range from 90° to 100°. Given these conditions, what is the correct range of measures possible for m∠C? 1) 20° to 40° 2) 30° to 50° 3) 80° to 90° 4) 120° to 130°

A

10. The degree measures of the angles of ΔABC are represented by x, 3x, and 5x - 24. Find the value of x.

ANS:

1. What is the measure of the largest angle in the accompanying triangle?

1) 41 2) 46.5 3) 56 4) 83

2. A billboard on level ground is supported by a brace, as shown in the accompanying diagram. The measure of angle A is 15° greater than twice the measure of angle B. Determine the measure of angle A and the measure of angle B.

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3. In right triangle ABC, m∠C = 3y – 10, m∠B = y + 40, and m∠A = 90. What type of right triangle is triangle ABC? 1) scalene 2) isosceles 3) equilateral 4) obtuse

4. If the measures of the angles of a triangle are represented by 2x, 3x – 15, and 7x - 15 the triangle is 1) an isosceles triangle 2) a right triangle 3) an acute triangle 4) an equiangular triangle

5. If the measures, in degrees, of the three angles of a triangle are x, x + 10 and

2x – 6,the triangle must be 1) isosceles 2) equilateral 3) right 4) scalene

ANS:

6. In ΔABC, the measure of m∠B is 21 less than four times the measure of m∠A, and the measure of m∠C is 1 more than five times the measure of m∠A. Find the measure, in degrees, of each angle of

1. The accompanying diagram shows the roof of a house that is in the shape of an isosceles triangle. The vertex angle formed at the peak of the roof is 84°.

What is the measure of x? 1) 138° 2) 96° 3) 84° 4) 48°

2. Tina wants to sew a piece of fabric into a scarf in the shape of an isosceles triangle, as shown in the accompanying diagram.

What are the values of x and y? 1) x = 42 and y = 96 2) x = 69 and y = 69 3) x = 90 and y = 48 4) x = 96 and y = 42

ANS:

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3. In the accompanying diagram, I isosceles ΔABC ≅ , isosceles ΔDEF, m∠C = 5x,

and m∠D = 2x + 18. Find m∠B and m∠BAG.

4. In the accompanying diagram, ΔABC and ΔABD are isosceles triangles with m∠CAB = 50 and m∠BDA = 55. If AB = AC and AB = BD, what is m∠CBD?

5. In isosceles triangle DOG, the measure of the vertex angle is three times the measure of one of the base angles. Which statement about ΔDOG is true? 1) ΔDOG is a scalene triangle. 2) ΔDOG is an acute triangle. 3) ΔDOG is a right triangle. 4) ΔDOG is a obtuse triangle

6. Vertex angle A of isosceles triangle ABC measures 20° more than three times m∠B. Find m∠C.

ANS:

7. Hersch says if a triangle is an obtuse triangle, then it cannot also be an isosceles triangle. Using a diagram, show that Hersch is incorrect, and indicate the measures of all the angles and sides to justify your answer.

8. If the vertex angles of two isosceles triangles are congruent, then the triangles must be 1) acute 2) congruent 3) right 4) similar

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9. In ΔABC, AB ≅ BC. An altitude is drawn from B to AC and intersects AC at D. Which conclusion is not always true? 1) ∠ABD ≅ ∠CBD 2) ∠BDA ≅ ∠BDC 3) AD≅BD 4) AD≅DC

10. In isosceles triangle ABC, AB = BC. Which statement will always be true? 1) m∠B = m∠A 2) m∠A > m∠B 3) m∠A = m∠C 4) m∠C < m∠B

13. If AB ≅ CD and CD ≅ EF and if AB = 2x + 3 and CD = 4x – 5, find EF.

14. Triangle ABC is congruent to triangle A`B`C`. If m∠C is represented by 2x – 10 and m∠C` is represented by x + 30: a) Find x b) Find the m∠C c) Find m∠B if it is represented by x - 25

15. Triangle DEF is congruent to triangle D`E`F`. If EF is represented by 3x + 2 and E`F` is represented by x + 10 and ED is represented by x + 2: a) find x, b) Find ED c) Find E`D` d) Find EF

16. Given that AD ≅ CB and ∠1 ≅ ∠2 and AB = 5x – 3, CD = 3x + 10 and BC = 2x + 5, write an equation to solve for x, and then find AB, CD, and BC.

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19. If ΔABC, BD is the median to side AC and must be congruent if ΔABD In ΔABC ΔCBD, then ΔABC must be 1) scalene 2) isosceles 3) right 4) equilateral

20. Two right triangles: 1) The hypotenuse of one triangle is congruent to the hypotenuse of the other. 2) An acute angle of one triangle is congruent to an acute triangle of the other. 3) Two leg of one triangle are congruent to two legs of the other. 4) Each contains a right angle.

21. Two isosceles triangles are congruent if 1) The vertex angle of one triangle is congruent to the vertex angle of the other. 2) A base angle of one triangle is congruent to a base angles of the other. 3) Leg of one triangle is congruent to a leg of the other 4) A leg and vertex angle of one triangle are congruent to a leg and vertex angle of the other.

22. In ΔABC, D is a point on BC such that AD is both angle bisector and an altitude in ΔABC. Which statement may be false? 1) BD = CD 2) AB = AC 3) AC = BC 4) m∠B = m∠C

23. In isosceles triangle ABC, ∠B is a right angle, AB = BC and BD is a median. Which statement is not an altitude of ABC? 1) BD 2) AD 3) AB 4) BC

24. In ΔABC, If CA ≅ CB and m∠A = 50, find the m∠B.