properties of angles and trianglescarussell.weebly.com/uploads/3/8/8/3/38837901/ch2_notes.pdf ·...
TRANSCRIPT
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Properties of Angles and Triangles
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In this unit, you will
● identify relationships among the measures of angles formed by parallel lines cut by a transversal.
● also prove deductively, the properties of angles in a triangle and other polygons.
● explore the different ways to prove that two triangles are congruent.
● then use this information to complete geometric proofs.
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Previous Knowledge
1. Sum of the angles of a triangle equal 180°– This will be proven later
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Previous Knowledge
2. Perpendicular and parallel lines– Perpendicular lines intersect at a right angle, 90°
– Parallel lines never intersect and are the same distance apart from each at all points
● Where could you see perpendicular lines in real life– Corners, desks, floors and walls
● Where could you see parallel lines in real life– Tiles, window with blinds, parking lot lines
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New Stuff
● Transversal– A line that crosses at least two other lines at
specific points
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● How many angles are formed when a transversal intersects two parallel lines?
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Properties of Angles
● Measure all the angles and note the patterns● Which angles are equal?● Which angles are supplementary?● Is this inductive reasoning?●
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Different types of Angles
● Vertically Opposite Angles– Angles that are opposite to each other when two
lines cross
– Vertically opposite angles are equal
– Angles a & b are vertically opposite angles
–
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● Corresponding Angles– When two lines are intersected by a transversal, the
angles in matching corners are corresponding angles
– Corresponding angles are equal
– a/e, b/f, c/g, d/h are corresponding angles
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● Alternate Interior Angles– Pairs of angles on opposite sides of the transversal,
but are inside the two lines
– Alternate interior angles are equal
– c/f and d/e are alternate interior angles
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● Alternate Exterior Angles– The pairs of angles on opposite sides of the
transversal, but outside the two lines
– Alternate exterior angles are equal
– a/h, b/g are alternate exterior angles
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● Consecutive Interior Angles– The pairs of angles on one side of the transversal,
but inside the two lines
– Consecutive Interior Angles are supplementary (add up to 180°)
– d/f, c/e are consecutive interior angles
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Note
● Consecutive Interior Angles are called interior angles on the same side of the transversal in the textbook
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How to tell if two lines are parallel
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Examples
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Summary of Angles
● Vertically Opposite Angles are equal● Corresponding Angles are equal● Alternate Interior/Exterior Angles are equal● Consecutive Interior Angles are supplementary
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● Name one pair of angles that are alternate exterior
● Name one pair of angles that are corresponding angles
● Name one pair of angles that are vertically opposite angles
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● s and z, t and y are alternate exterior● s and w, u and y, t and x, v and z are
corresponding angles● s and v, t and u, w and z, x and y are vertically
opposite angles
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● Find the measure of ALL the missing angles
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● Are these lines parallel?
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● No they are not b/c the angles shown are corresponding angles and they are not equal
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● Are these lines parallel?
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● Yes there are b/c the angles shown are consecutive interior angles and they are supplementary (they add up to 180 degrees)
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● Are these lines parallel?
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● No they are not b/c alternate exterior angles are not equal
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For you to do!
● Pg. 72 #'s 5, 6
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Notes on Notation● When writing angles we sometimes use 3 points,
with the middle point being the vertex of the angle● An angle symbol, , is placed in front of the 3
points● AXZ or ZXA NOT ZAX
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● On occasion, we use only the point at the vertex if there will be no confusion as to what angle we are looking at
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● For parallel lines, when we say line AB is parallel to line CD, we use two vertical lines to indicate they are parallel
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Section 2.2: Angles formed by parallel lines
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Drawing parallel lines using a compass
● Draw a straight line and a point (P) NOT on the line
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● Draw a line through P intersecting the line at a point (Q)– The exact angle is not important
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● Using a compass, construct an arc centered at Q and passing through both lines. Label the intersection points R and S.
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● Draw another arc, centered at P, with the same radius as the arc RS. Label the intersection point (T).
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● Draw a third arc, with center T, and same radius as RS that intersects the previous arc drawn. Label the intersection point W.
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● Draw a line that goes through P and W.
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● To show that lines PW and QS are parallel, just show that one of the following angle properties is true.
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● Find the missing angles. Justify your answers.
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● x = 62 (corresponding angles)● y = 118 (supplementary angles)● z = 62 (vertically opposite angles with x)
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● Find the missing angles. Justify your answer.
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● x = 125 (supplementary angles)● y = 55 (alternate interior angles or consecutive
interior angles are supplementary)● Z = 125 (corresponding angles)
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● A = 110 (corresponding angle)● B = 110 (vertically opposite angle with a)● C = 70 (consecutive interior angles are
supplementary)● D = 70 (alternate interior angles)
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● Determine the value of x
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● They are alternate exterior angles and thus equal– 2x – 10 = x + 15
– x = 25
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● Fix the errors in the following problem
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● Here's what it should look like.
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Proving conjectures using two-column proofs
● Conjecture:
When a transversal intersects a pair of parallel lines, the alternate interior angles are equal.
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● Conjecture:– When a transversal intersects a pair of parallel
lines, the alternate exterior angles are equal.
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Example
● Determine the missing angles
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● Conjecture:– When a transversal intersects a pair of parallel
lines, the interior angles on the same side as the transversal are supplementary
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● Given the situation below, would you expect Prince Phillip Drive and Elizabeth Avenue to ever cross if they went on forever?
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For you to do!
● Pg.78-82 #'s: 1, 2, 3, 4, 15, 20
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Section 2.3: Angles Properties in Triangles
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Proving the sum of interior angles of triangles is 180°
● Draw a triangle● Draw a line which is parallel to one of the sides
of the triangle and tangent at one of the vertices
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● Identify pairs of equal angles
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● What is the sum of the measures of <1, <2, <3?● <1 + <2 + <3 = 180° (Straight line)● Prove <2 + <4+ <5 = 180°
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● Determine the measure of <P
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● (15x - 4) + (4x + 5) + 65 = 180● 19x + 66 = 180● 19x = 114● X = 6
● <P– 15(6) – 4 = 86°
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Exterior Angles
● In the diagram, <MTH is an exterior angle of ΔMAT
● Determine the missing angles
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● <T = 180 – 155 = 25°● <A = 180 – 25 – 40 = 115°
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Note
● The measure of any exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles
● <d = <a + <b
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Examples
● What is measure of <XYC?
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● Find the measures of a & b
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Section 2.4: Angle Properties in Polygons
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● Convex polygon – Each interior angle is less than 180°
● Concave polygon– One or more interior angle(s) is greater than 180°
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How is the number of sides in a polygon related to the sum of its
interior angles?
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Polygon Number of sides (n)
Number of triangles
Angle Sum (S)
Triangle 3 1 1(180) = 180
Quadrilateral 4 2 2(180) = 360
Pentagon 5 3 3(180) = 540
Hexagon 6 4 4(180) = 720
Heptagon 7 5 5(180) = 900
... ... ... ...
decagon 10 8 8(180) = 1440
100-gon 100 ? ?
n-gon n n-2 (n-2) x 180
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The Sum of the Interior Angles of a Polygon:
S = (n-2) x 180°● Where n = number of sides
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Examples● Determine the sum of the interior angles in the
following polygons● A) B)
A) n = 8 B) n = 12
S = 180(8-2) S = 180(12-2)
S = 180(6) S = 180(10)
S = 1080° S = 1800°
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● Determine the number of sides of a polygon if the sum of its interior angles is 1620°
S = 180(n-2)
1620 = 180(n-2)
9 = n-2
n = 11
The polygon has 11 sides
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● Here's home plate. It has 3 right angles and two congruent angles (A & B). What are they?
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● Find the sum of its interior angles
S = 180(5-2) = 540°
● Subtract from the total the right angles
540 – 3(90) = 270°
● Since A = B, half of 270 is their angle
270 = 135°
2
A = B = 135°
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● Regular polygon– All sides are the same length and all interior angles
are equal
– If this isn't the case, it's an irregular polygon
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Examples
● What is the measure of each interior angle of the following regular polygons?
A) B)
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● A) B)
S = 180(4-2) S = 180(5-2)
S = 360° S = 540°
360 = 90° 540 = 108°
4 5
Each interior Each interior angle
angle is 90° is 108°
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● Find the measures of x and y
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● S = 180(10-2) (Find the total angle sum)● S = 1440°● 1440 = 144° (Find each interior angle)
10● 180 – 144 = 36° (Find the sum of x and y)● 36 = 18° = x = y (Find x and y)
2● x and y equal 36°
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Summary of results
● Interior angle sum of a convex polygon– (n – 2) x 180
● Interior angle of a convex polygon– (n-2) x 180
n
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Section 2.5: Exploring Congruent Triangles
Section 2.5 – 2.6: Exploring Congruent Triangles
and proving triangles are
congruent
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The idea of congruence
● Two geometric figures with exactly the same size and shape
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Congruent or similar?
● The two shapes need to be the same size to be congruent
● When you need to alter one shape to make it the same as the other, then those shapes are similar
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Congruent?
● Why such a funny word that basically means “equal”?– They would only be “equal” if laid on top of each
other
– Comes from the Latin word congruere which means “to agree”
– So shapes “agree” when they are congruent
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Congruent Triangles
● If two triangles are congruent, they will have exactly the same three sides and angles
● They may not be in the same position, but they will be there
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Same sides
● If all 3 sides are the same, then triangles are congruent
● For example:
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But...
● The following triangles are not congruent b/c they do not have the same sides.
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Same Angles
● Doesn't always work with angles● Two triangles can have the same angles, but be
different sizes– All the angles can match, but sides are different
lengths
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They could be congruent if they are the same size
● In this case, they happen to be the same size● Having the same angle is no guarantee
triangles are congruent
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Definition of Congruent Triangles
● Two triangles are congruent if and only if (IFF) their corresponding parts are congruent
● CPCTC● Corresponding Parts of Congruent Triangles
are Congruent
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Corresponding Parts
● Side-Side-Side● Angle-Side-Angle● Side-Angle-Side● Angle-Angle-Side
● To prove two triangles are congruent, one only needs to prove one of the above relationships is true
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Side-Side-Side Congruence
● If the sides of one triangle are the same as the sides of another triangle, then the triangles are congruent
http://www.mathopenref.com/congruentsss.html
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Angle-Side-Angle Congruence● If two angles and the included side of one triangle
are the same as two angles and the included side of another triangle, then the triangles are congruent
● These triangles are congruent
http://www.mathopenref.com/congruentasa.html
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Side-Angle-Side Congruence● If two sides and the included angle of a triangle are
the same as two sides and the included angle of another triangle, then the triangles are congruent
● The following triangles are congruent
http://www.mathopenref.com/congruentsas.html
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Angle-Angle-Side Congruence● If two angles and the non-included side are the same
as two angles and the non-included side of another triangle, then the triangles are congruent
● The following triangles are congruent
http://www.mathopenref.com/congruentaas.html
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The Congruence Postulates
● Side-Side-Side (SSS)● Angle-Side-Angle (ASA)● Side-Angle-Side (SAS)● Angle-Angle-Side (AAS)
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Name that Pokemon...I mean Postulate!
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For you to do
● Pg. 106 #'s 1,2,3,4
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2.6: Proving congruent triangles
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Steps for proving triangles are congruent
● 1. Mark the givens● 2. Mark any reflexive sides, vertical angles,...● 3. Choose a method (SSS, SAS, ASA, AAS)● 4. List the parts... in the order of the method● 5. Fill in the reasons...why you marked the parts
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● Pg. 113 #'s 1,2
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That's all folks!