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Name:
Unit 4 Congruency and Triangle Proofs
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Triangle Congruence and Rigid Transformations In the diagram at the right, a transformation has occurred on ABC. Describe a transformation that created image ABC from ABC. Is ABC congruent to ABC? Explain. The vertices of MAP are M(-8, 4), A(-6, 8) and P(-2, 7). The vertices of MAP are M(8, -4), A(6, -8) and P(2, -7). Plot MAP. Verify that the sides of the triangles are congruent. Describe a rigid motion that can be used to MAP Given PQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent to STR with S(2, -4), T(6, 2) and R(0, 0). Plot STR. Describe a rigid motion which can be used to verify the triangles are congruent.
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Given RST with R(1, 1), S(4, 5) and T(7, 5). Plot the reflection of RST in the y-axis and label it RST. Is RST congruent to RST? Explain. Plot the image of RSTunder the translation (x, y) (x + 4, y β 8). Label the image of RST. Is RST congruent to RST? Explain. Is RST congruent to RST? Explain. Given DFE with D(1, -1), F(9, 6) and E(5,7) and BAT with B(1, 1), A(-6, 9) and T(-7, 5). Describe a transformation that will yield BAT as the image of DFE. Is BAT congruent to DFE? Explain. Given CAP with C(-4, -2), A(2, 4) and P(4, 0) and SUN with S(-8, -4), U(4, 8) and N(8, 0). Describe a transformation that will yield SUN as the image of CAP. c) Is CAP congruent to SUN? Explain.
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Discovering Congruent Triangles Activity
Part 1 1. Have students put the 3 straws of different lengths together
to form a triangle as shown. 2. Form another triangle with the other set of straws. 3. Measure the angles of both triangles using a protractor. Questions:
1. What are the measures of the 3 angles in the first triangle? 2. What are the measures of the 3 angles in the second triangle? 3. What is the relationship between the angles of each triangle? 4. Are the triangles congruent? 5. Can the straws be rearranged to form a triangle with different angles?
Part 2 1. Take 2 of the straws, place them on a piece of paper, and form a 60
degree angle between them. 2. Take the 2 straws of the same length and also form a 60 degree angle
between them. 3. Draw a line to represent the 3rd side. Repeat the process for the 2nd
triangle. 4. Measure the length of the 3rd side and the two remaining angles for each
triangle. Questions:
1. What is the length of the 3rd side? 2. What are the measures of the remaining angles? 3. Are the two triangles congruent? 4. Use any two straws and any angle of your choice. Do you get the same result? Will you always get the same result?
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Part 3 1. Measure three angles measuring 80, 60, and 40 degrees on the corners
of 2 pieces of construction paper or cardstock, cut them out, and label them.
2. On a piece of paper, take one of the straws, and place two of the cut-out angles on each end as shown. Repeat the process for the 2nd triangle.
3. Using a ruler, draw a segment along each of the angle. The two segments should intersect forming the last angle. Repeat the process for the 2nd triangle.
4. Measure the 3rd angle and the lengths of the 2 sides in each triangle. Questions:
1. What is the measure of the 3rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent? 4. What if you used the 5cm straw? The 8cm straw? A straw with a different length?
Part 4
1. Use two of the angles used in the example above. 2. Use one of the straws and place one of the angles alongside it as
shown. Draw a long segment like the dashed one in the drawing. Repeat the process for the 2nd triangle.
3. Place the second angle along this segment so that when a 2nd segment is drawn, it will connect with the end of the straw.
4. Measure the 3rd angle and the two remaining sides. Questions:
1. What is the measure of the 3rd angle for each triangle? 2. What are the measures of the remaining 2 sides for each triangle? 3. Are the triangles congruent?
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Part 5
1. Place two of the straws together forming an angle of any degree for one triangle, and repeat the process for the 2nd triangle.
2. Use one of the pre-cut angles and place alongside the longer of the sides but not as the included angle.
3. Draw a segment to connect the 3rd side to the other two sides.
4. Swing the 8cm straw so that it hits the 3rd side at a different spot in the 2nd triangle as in the first.
5. Measure the 3rd side and the remaining 2 angles in each triangle.
Questions
1. What is the measure of the 3rd side for each triangle? 2. What are the measures of the remaining 2 angles for each triangle? 3. Are the two triangles congruent? 4. Do you think that you would get different results if you used a different angle?
Part 6
1. Place the 3 angles so that they can form a triangle without measuring the sides initially. Draw segments connecting the angles. Repeat the process for the second triangle.
2. Measure the 3 sides for each triangle. Questions
1. What are the measures of the 3 sides for each triangle? 2. Are the two triangles congruent?
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3 cm2 cm
4 cm4 cm
3 cm2 cm
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Congruent Triangles Investigation Part I What does it mean to say two triangles are congruent? List the ways to justify that triangles are similar. Examine the triangles with all side lengths labeled. Are they similar? Why? What is the scale factor? _______ : _______ What do we know about the corresponding angles of similar triangles? What does this tell us about the pair of triangles? Part 2: Examine the triangles with two sides lengths and an included angle labeled. Are they similar? Why? What is the scale factor? _______ : _______ Since the triangles are similar, what do we know about P and H? What do we know about I and O? Use the scale factor you gave in part b to determine the length of ππ»Μ Μ Μ Μ Μ . What does this tell us about the pair of triangles? Part 3: Examine the triangles with two angle pairs marked congruent. Are they similar? Why? What is the scale factor? _______ : _______ Since the triangles are similar, what do we now bout K and Y? Use the scale factor you gave in part c to determine the lengths of πΎπΏΜ Μ Μ Μ and ππΜ Μ Μ Μ . What does this tell us about the pair of triangles?
4 cm
2.15 cm
4 cm
3.2 cm
32523252
K
L M ZX
Y
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Think back to the three situations we examined. In #1, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle. The triangles are congruent by the
___________, __________, __________ (SSS) Postulate. In #2, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of sides and an included angle of another triangle. The triangles are congruent by the
___________, __________, __________ (SAS) Postulate. In #3, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of angles and an included side of another triangle. The triangles are congruent by the
___________, __________, __________ (ASA) Postulate. Two other Postulates
Angle, Angle, Side Postulate (AAS Theorem) How can you change this into one of the Postulates that we already have?
Hypotenuse, Leg Theorem (HL Theorem) By Pythagorean Theorem, if you know the hypotenuse and 1 leg, you can calculate the 2nd leg by _________________ Theorem and prove congruence by ____________ postulate. Note: Postulate β Statement which is taken to be true without proof. Theorem β Statement that can be demonstrated to be true by accepted mathematical operations and arguments.
3 cm2 cm
4 cm4 cm
3 cm2 cm
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1.5 cm
5 cm
4 cm
4 cm
5 cm H
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4 cm
2.15 cm
4 cm
3.2 cm
32523252
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L M ZX
Y
5 cm 5 cm
10 cm 10 cm
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HW: How do you prove triangles are congruent? #1 & 2 Use the given coordinates to determine if ABC DEF
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CPCTC Essential question: What can you conclude about two triangles that are congruent? When you know that two triangles are congruent, you can make conclusions about the sides and angles of the triangles.
Reflect: If you know that ABC DEF, what six congruent statement about segments and angles can you write? Why?
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When two triangles are congruent the corresponding parts are the sides and angles that are images of each other. You write congruence statements for two figures by matching the corresponding parts. In other words, the statement ABC DEF contains the information that π΄π΅Μ Μ Μ Μ corresponds to π·πΈΜ Μ Μ Μ so that π΄π΅Μ Μ Μ Μ β π·πΈΜ Μ Μ Μ , A corresponds to D so that A D, and so on.
Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC)
If two triangles are congruent, then the corresponding sides are congruent and the corresponding angles are congruent.
Converse of the Corresponding Parts of Congruent Triangles are Congruent Theorem
(CPCTC)
If tow triangles have corresponding sides that are congruent and the corresponding angles that are congruent, then the triangles are congruent.
Examples
Discuss:
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CPCTC and Naming Congruent Triangles I. Draw and label a diagram. Then solve for the variable and the missing measure or length. 1. If βπ΅π΄π β βπ·ππΊ, and πβ π΅ = 14, πβ πΊ = 29, πππ πβ π = 10π₯ + 7. Find the value of x πβ π.
x = ___________ πβ π= _________ 2. If βπΆππ β βππΌπΊ, and πΆπ = 25, πΆπ = 18, πΌπΊ = 23, πππ ππΊ = 7π₯ β 17 . Find the value of x and PG.
x = ___________
PG=___________ 3. If βπ·πΈπΉ β βπππ and π·πΈ = 3π₯ β 10, ππ = 4π₯ β 23, πππ ππ = 2π₯ + 7. Find the value of x and EF.
x = ___________ EF = __________ II. Use the given information and triangle congruence statement to complete the following. 4. βπ΄π΅πΆ β βπΊπΈπ, AB = 4, BC = 6, and AC = 8. What is the length of πΊπΜ Μ Μ Μ ? How do you know?
5. βπ΅π΄π· β βπΏππΎ, πβ π· = 52Β°, πβ π΅ = 48Β°, πππ πβ π΄ = 80Β°.
a. What is the largest angle of βπΏππΎ?
b. What is the smallest angle of βπΏππΎ?
6. βπππ β βπ»ππ. βπππ is isosceles. Is there enough information to determine if βπ»ππ is isosceles?
Explain why or why not.
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III. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write βnoneβ in the blanks.
7. βπΈπΉπ· β β___________ 8. βπ΄π΅πΆ β β___________ 9. βπΏπΎπ β β___________ by ________ by ________ by ________
10. βπ΄π΅πΆ β β___________ 11. βπ΄π΅πΆ β β___________
by ________ by ________ IV. Use the given information to mark the diagram and any additional congruence you can determine from the diagram. Then complete the triangle congruence statement and give the reason for triangle congruence . 12. 13. Given: β 1 β β 3, β 2 β β 4 Given: β π΄π΅π· β β πΆπ΅π·, β π΄π·π΅ β β πΆπ·π΅ βπ΄π΅πΆ β β__________ by __________ βπ΄π΅π· β β__________ by __________ 14. 15. Given: πΊ ππ π‘βπ ππππππππ‘ ππ πΉπ΅Μ Μ Μ Μ πππ πΈπ΄Μ Μ Μ Μ Given: β 1 β β 3, πΆπ·Μ Μ Μ Μ β π΄π΅Μ Μ Μ Μ βπ΄π΅πΊ β β__________ by __________ βπ΄π΅πΆ β β__________ by __________
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Congruent Triangle Problems - Honors I. Ξπππ β Ξπ΄π΅πΆ. Find the values of x and y.
1. πβ π = 5π₯ + 70, πβ πΆ = 24π₯ β 25, ππ = 4π¦ + 2, π΅πΆ = π₯ + π¦
2. πβ π = 90 β π¦, πβ πΆ = 13, ππ = 3π₯ + π¦ β 1, π΄πΆ = 32 β π₯
3. ππ = 5π₯ β 31, ππ = β3π¦ β 1, π΅πΆ = π₯ + 1, π΄π΅ = 9 β π¦
4. πβ π΄ = 15π¦ β 3, πβ π = 43 β π₯, ππ = 11 β π₯, π΄π΅ = 3π¦ + 1
5. π΄π΅ = 2π₯ + π¦, ππ = 7, π΅πΆ = 11, ππ = 4π₯ + π¦
6. Ξπππ β Ξπππ, πβ π = π₯ + 10, πβ π = π¦ + 20, πβ π = 3π₯, and πβ π = π₯ + 3π¦. Find πβ π and
πβ π.
II. Indicate which triangles are congruent. Be sure to have the correspondence of the letters correct.
a. ΞπΈπ πΆ β _______ b. E is the midpoint of ππΜ Μ Μ Μ c. Ξπ΅ππ β _______
Why is π πΆΜ Μ Μ Μ β π πΆΜ Μ Μ Μ ? ΞπππΈ β _______ Why is β 1 β β 2?
E C
TR
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III. Coordinate Geometry
1. Graph each line on a coordinate plane. Identify two congruent triangle formed by the lines. Explain why
the triangles are congruent.
x=0, y = 0, x = 4, y = 2x β 4
2. Consider two triangles, Ξπ΄π΅πΆ and ΞπΉπ·πΈ, with vertices A = (0, 7), B = (-4, 0), C = (0, 0), D = (2, 3),
E = (2, -1), and F = (9, -1). Draw a diagram and explain why Ξπ΄π΅πΆ β ΞπΉπ·πΈ.
IV. Solve.
1. The perimeter of ABCD is 85. Find the value of x. Is Ξπ΄π΅πΆ congruent to Ξπ΄π·πΆ? Explain.
2. Given: ΞππΈπ β ΞπΆπ΄π
EN = 11
AR = 2x β 4y
NW = x + y
CA = 4x + y
EW = 10
Draw the triangles, solve for x and y, and find CR.
3x + 4 5x - 7
6x - 114x
C
A
B D
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DA C
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Introduction to Triangle β Proof Ex 1) Given: π΄π·Μ Μ Μ Μ β π·πΆΜ Μ Μ Μ
π΄πΆΜ Μ Μ Μ β₯ π΅π·Μ Μ Μ Μ Prove: ΞABD β ΞCBD Given Given Reflexive Prop β Ex 2) Given: <E β <H G is the midpoint of πΈπ»Μ Μ Μ Μ Prove: ΞGFE β ΞGIH
Ex 3) Given: π½πΎΜ Μ Μ // ππΏΜ Μ Μ Μ
π½πΎΜ Μ Μ β ππΏΜ Μ Μ Μ Prove: β‘π½ β β‘πΏ
G
F
E
H
I
G is the midpoint of πΈπ»Μ Μ Μ Μ
<FGE β <IGH
π½πΎΜ Μ Μ // ππΏΜ Μ Μ Μ π½πΎΜ Μ Μ β ππΏΜ Μ Μ Μ
Ξ β Ξ
Reflexive Prop β
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Ex 4) Given: π΄π΅Μ Μ Μ Μ β π΅πΆΜ Μ Μ Μ π΄πΆΜ Μ Μ Μ β₯ π΅π·Μ Μ Μ Μ Prove: ΞABD β ΞCBD Use separate paper to complete the following.
Ex 5) Given: π΄π΅Μ Μ Μ Μ // πΆπ·Μ Μ Μ Μ
π΄π΅Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ Prove: π΄πΈΜ Μ Μ Μ β πΈπΆΜ Μ Μ Μ Ex 6) Given: πΌπΊΜ Μ Μ bisects < FIJ πΌπΉΜ Μ Μ β πΌπ»Μ Μ Μ Μ Prove: β‘F β β‘H Ex 7) Given: πΎπΏΜ Μ Μ Μ // π½πΜ Μ Μ Μ πΎπ½Μ Μ Μ // πΏπΜ Μ Μ Μ Prove: πΎπ½Μ Μ Μ β πΏπΜ Μ Μ Μ
E
A
B
D
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F HG
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JM
L
Reflexive Prop β
Def of right Ξ
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Triangle Proofs Practice Complete the following proofs using separate paper. Draw and mark each picture before writing the proof.
1. Given: BDβ₯AC
ADΜ Μ Μ Μ β DCΜ Μ Μ Μ
Prove: β ABDβ β CBD
2. Given: G is the midpoint of FH
πΈπΉΜ Μ Μ Μ β πΏπ»Μ Μ Μ Μ
β Eβ β L
Prove: EGβ LG
3. Given: πΆπ·Μ Μ Μ Μ bisects β π΄πΆπ΅
β π΄ β β π΅
Prove: π΄π·Μ Μ Μ Μ β π·π΅Μ Μ Μ Μ
4. Given: BDβ₯AC
ABΜ Μ Μ Μ β BCΜ Μ Μ Μ
Prove: β ABDβ β CBD
5. Given: ππ Μ Μ Μ Μ β ππΜ Μ Μ Μ
β‘Pβ β‘S
β‘Tβ V
Prove: ππ Μ Μ Μ Μ β ππΜ Μ Μ Μ
D
C
A B
P
S
T
V
R
Q
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6. Given: π΅π·Μ Μ Μ Μ bisects β π΄π΅πΆ
π΅π΄Μ Μ Μ Μ β πΆπ΅Μ Μ Μ Μ
Prove: β‘ADB β β‘BDC
7. Given: G is the midpoint of FIΜ β‘F β β‘I
Prove: πΈπΉΜ Μ Μ Μ β πΌπ»Μ Μ Μ Μ
8. Given: πΈπΉΜ Μ Μ Μ //π»πΌΜ Μ Μ Μ G is the midpoint of πΈπ»Μ Μ Μ Μ
Prove: πΉπΊΜ Μ Μ Μ β πΊπ»Μ Μ Μ Μ
9. Given: π½πΜ Μ Μ Μ //πΏπΎΜ Μ Μ Μ β‘J β β‘L
Prove: π½πΎΜ Μ Μ β ππΏΜ Μ Μ Μ
10. Given: π½πΜ Μ Μ Μ //πΏπΎΜ Μ Μ Μ π½πΎΜ Μ Μ //πΏπΜ Μ Μ Μ
Prove: β‘J β β‘L
B
D
A C
G
F
E
H
I
G
F
E
H
I
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Q
M P
R
N
A B
D C
2
1
F
IH
GK
J
MORE PRACTICE WITH PROOF EX 1) GIVEN: ππΜ Μ Μ Μ Μ β ππ Μ Μ Μ Μ , <M AND <P ARE RIGHT ANGLES. N IS THE MIDPOINT OF ππΜ Μ Μ Μ Μ PROVE: <MQN β <PRN
EX 2) GIVEN: π΄π·Μ Μ Μ Μ β π΅πΆΜ Μ Μ Μ <ADC β < BCD
PROVE: π΄πΆΜ Μ Μ Μ β π΅π·Μ Μ Μ Μ
EX 3) GIVEN: <I β <G <1 β <2
π½οΏ½Μ οΏ½ β πΎπΊΜ Μ Μ Μ PROVE: <1 β < 2
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EX 4) GIVEN: π΄π΅Μ Μ Μ Μ // πΆπ·Μ Μ Μ Μ π΄π΅Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ <AEB β <DFC
PROVE: π΅πΈΜ Μ Μ Μ β π·πΉΜ Μ Μ Μ
EX 5) GIVEN: ππΌΜ Μ Μ β π·πΌΜ Μ Μ π πΌΜ Μ Μ β πΈπΌΜ Μ Μ PROVE: β π β β πΈ
EX 6) GIVEN: πΎπΜ Μ Μ Μ Μ β₯ π½οΏ½Μ οΏ½ π IS THE MIDPOINT OF π½οΏ½Μ οΏ½ PROVE: βπ½πΎπ β βπΏπΎπ
P
E
D
I
R
2
1
D
B C
A
E
F
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E
FH
I
G
Unit 4B Day 4: More Practice with Proof
1. Given: π΄π΅Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
π΄π΅Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ
πΆπ·Μ Μ Μ Μ β₯ π΅πΆΜ Μ Μ Μ
π΄πΈΜ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
Prove: β‘A β β‘D
2. Given: πΊπΎΜ Μ Μ Μ β π»πΏΜ Μ Μ Μ
πΊπΏΜ Μ Μ Μ β π»πΎΜ Μ Μ Μ
Prove: β‘K β β‘L
3. Given: π΄πΆΜ Μ Μ Μ β π΅πΆΜ Μ Μ Μ
π΄πΈΜ Μ Μ Μ β π΅π·Μ Μ Μ Μ
Prove: πΆπ·Μ Μ Μ Μ β πΆπΈΜ Μ Μ Μ
4. Given: <F and <H are right angles
G is the midpoint of πΉπ»Μ Μ Μ Μ
πΈπΊΜ Μ Μ Μ β πΏπΊΜ Μ Μ Μ
Prove: β‘E β β‘L
5. Given: β‘1 β β‘2
β‘B β β‘ECF
π΅π·Μ Μ Μ Μ β πΆπΉΜ Μ Μ Μ
Prove: π΄π·Μ Μ Μ Μ β πΈπΉΜ Μ Μ Μ
B
C
A
D
E
F
G H
K L
C
A B
ED
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A
B F
E
C D
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6. Given: β‘B β β‘C
π΅πΉΜ Μ Μ Μ β πΊπΆΜ Μ Μ Μ
π΅π·Μ Μ Μ Μ β πΈπΆΜ Μ Μ Μ
Prove: β‘BDF β β‘CEG
7. Given: π΄π΅Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
π΄π΅Μ Μ Μ Μ // πΆπ·Μ Μ Μ Μ
π΄πΈΜ Μ Μ Μ β πΆπΉΜ Μ Μ Μ
Prove: π΅πΈΜ Μ Μ Μ β π·πΉΜ Μ Μ Μ
8. Given: β π·π΄πΏ β β π΅πΆπ
π·πΏΜ Μ Μ Μ β ππ΅Μ Μ Μ Μ Μ
β‘ALD and CMB are right angles
Prove: π΄πΏΜ Μ Μ Μ β πΆπΜ Μ Μ Μ Μ
9. Given: πΉπΌΜ Μ Μ πππ πππ‘π πΈπ»Μ Μ Μ Μ
β‘E β β‘H
Prove: πΈπΉΜ Μ Μ Μ β π»πΌΜ Μ Μ Μ
10. Given: πΉπΌΜ Μ Μ πππ π»πΈΜ Μ Μ Μ πππ πππ‘ πππβ ππ‘βππ
Prove: β‘E β β‘H
B C
A
F
DE
G
2
1
D
B C
A
E
F
2
1
B
D C
A
L
M
G
F
E
H
I
G
F
E
H
I
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Isosceles Triangle investigation
1. In the box, draw an angle
and label the vertex C. This
will be your vertex angle.
Measure C.
2. Using point C as center,
swing an arc that intersects
both sides of C
3. Label the points of
intersection A and B.
Construct side AB. You
have constructed isosceles
ΞABC with base AB.
4. Measure sides AC and BC. What is the relationship between AC and BC?
5. Use your protractor to measure the base angles (A and B) of isosceles ΞABC.
6. Compare your results with the rest of the class. What relationship do you notice about the
base angles of each isosceles triangle?
Isosceles Triangle Theorem: If 2 sides of a triangle are congruent, then
_______________________________________________________________________________.
Isosceles Triangle Theorem Converse: If 2 angles of a triangle are congruent, then
______________________________________________________________________________.
1. If πΆπΜ Μ Μ Μ Μ β πΈπΜ Μ Μ Μ Μ , then _________ β _________ by
_____________________________________________.
2. If β 2 β β 3, then _________ β _________ by
_____________________________________________.
431 2
OC E
M
R
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Isosceles Triangle Practice 1. In triangle ABC, AB CB and mCBD = 124. Find the measure of A.
2. In isosceles triangle ABC, AB = AC. mC = 6x + 10 and mB = 3x + 40. Find the measure of the exterior angle at the vertex angle A.
3. Find the value of x: a.
b.
c.
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5
43Y
X
Z
1 23 4
ZK
V
LO
21
P
A DB C
Practice with Isosceles Triangle Theorem and Converse Proofs
1. Given: ππΜ Μ Μ Μ β ππΜ Μ Μ Μ
Prove: β‘3 β β‘5
2. Given: πΎπΜ Μ Μ Μ β ππΜ Μ Μ Μ
πΎπΜ Μ Μ Μ β πΏπΜ Μ Μ Μ
Prove: ΞKVO β ΞZVO
3. Given: β‘1 β β‘2
π΄π΅Μ Μ Μ Μ β πΆπ·Μ Μ Μ Μ
π΄πΜ Μ Μ Μ β ππ·Μ Μ Μ Μ
Prove: ΞABP β ΞDCP
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T
P RI
AE
21G
J
K
M
50
x
x 54
63
11
10x
2x + 75x - 8
4040
F G
4x - 6
18
16
F
G H
x
98
x
74
12
10
10
A D
E
B C
Find the value of the variable or question mark. 1. 2. 3. 4. x = ________ x = ________ x = ________ x = ________ 5. 6. 7. 8. x = ________ x = ________ x = ________ x = ________ Complete the following using the diagram to the right. 9. a. If πΈπ΄Μ Μ Μ Μ β πΈπ·Μ Μ Μ Μ , then _______ _______.
b. If πΈπ΅Μ Μ Μ Μ β πΈπΆΜ Μ Μ Μ , then _______ _______.
c. If ΞEAD is an isosceles right triangle with right angle AED,
then the measure of A is ________.
10. Given: ππΜ Μ Μ Μ β ππ Μ Μ Μ Μ
πΈπΜ Μ Μ Μ β π΄π Μ Μ Μ Μ
I is the midpoint of ππ Μ Μ Μ Μ
Prove: πΈπΌΜ Μ Μ β π΄πΌΜ Μ Μ
11. Given: M is the midpoint of π½πΎΜ Μ Μ
β 1β β 2
Prove: JG β MK
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DB C
A
Use your knowledge of congruent triangle proofs to complete the following flow proofs. 1. Given: π΄π· β‘ ππ π‘βπ β₯ πππ πππ‘ππ ππ π΅πΆΜ Μ Μ Μ
Prove: π΄π΅ = π΄πΆ
π΄π· β‘ ππ π‘βπ β₯ πππ πππ‘ππ ππ π΅πΆΜ Μ Μ Μ
π΄π· β‘ β₯ π΅πΆΜ Μ Μ Μ
π· ππ π‘βπ ππππππππ‘ ππ π΅πΆΜ Μ Μ Μ
Reflexive prop of β
β π΄π·π΅ β β π΄π·πΆ βπ΄π΅π·β βπ΄πΆπ·
π΄π΅ = π΄πΆ
Theorem: If a point is on the perpendicular bisect of a segment,
it is __________________________________ from the _________________________
of the segment.
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4
3
2
1
E
FB
D
A
C
Given: π· ππ ππ π‘βπ πππ πππ‘ππ ππ β π΄π΅πΆ
π·πΈΜ Μ Μ Μ β₯ π΅π΄ , π·πΉΜ Μ Μ Μ β₯ π΅πΆ
Prove: π·πΈ = π·πΉ
π· ππ ππ π‘βπ πππ πππ‘ππ ππ β π΄π΅πΆ
π·πΈΜ Μ Μ Μ β₯ π΅π΄
π·πΉΜ Μ Μ Μ β₯ π΅πΆ
reflexive prop of β
Theorem: If a point is on the bisector of an angle, it is __________________________________
from the _________________________ of the angle.
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BC D
A
P
NM
Q
L
O
PH
I
G
T
S
U
R
B
D
C
A
E
W
VK
F
J
A B
P C
Q
Letβs Make a Proof! You have spent lots of time in this unit writing proofs that someone else designed. Now it is your turn to make your own! Use the guidelines below to make up and do at least 2 proofs. Each of your 2 proofs should consist of: One of the given diagrams
The appropriate βgivenβ and βproveβ statements to set up your proof
The correctly completed proof using the diagram and βgivenβ and βproveβ statement you wrote
At least 3 vocabulary words from the given list (within the completed proof)
At least 3 of the rules from the postulates, property, and theorems list (within the completed
proof)
The second proof you write must use a different diagram, different vocabulary words, and different rules from the first one to meet the minimum requirements.
Diagram Choices: 1. 2. 3. 4. 5. 6. 7.
Vocabulary Terms: o Congruent segments
o Congruent angles
o Midpoint
o Segment bisector
o Angle bisector
o Perpendicular lines
o Perpendicular bisector
o Right angle
o Right triangle
Rules (Postulates, Properties, Theorems): o Vertical angles are congruent.
o Reflexive property of congruence
o All right angles are congruent.
o If β₯ lines are cut by a transv., corr. β π are β .
o If β₯ lines are cut by a transv., alt. int. β π are β .
o If β₯ lines are cut by a transv., alt. ext. β π are β .
o SSS
o SAS
o ASA
o AAS
o HL
o CPCTC
o Isosceles Triangle Theorem
o Isosceles Triangle Theorem Converse
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Test Review 1. Given BED BUG, find x and BU if BE = 6x+16, UG=10x-2 and ED = 6x+6.
x=__________
BU=________________
2. In βABC, AB, AC = 6x-5, BC = 3x+13, and AB = 4x+7. Find x and the length of the base.
x
___________
base =
___________
3. In an isosceles triangle, a vertex angle measures 36β°. What is the measure of each base angle? 4. Decide whether it is possible to prove the triangles are congruent. If yes, then state which
congruence postulate you would use.
a. b. c. d. 5. If ABCPQR and C=2x+2, R=3x-18, find the value of x. 6. βXYZ βJKL. IfY=14-x, K=2x+50 and L=-4x find the Zm . 7. IfABCLMN, and AB=4x-y, LM =2x-2y, BC = x-3y and MN =21 find the value of x and y.
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8. Suppose βABC βEFG. For each of the following, name the corresponding part.
A : ______________ BCA : ______________ AC : _______________
9.
10. In the given triangles, ABCXYZ Which two statements identify corresponding congruent parts for these triangles?
A. B XY and CY
B. AB YZ and CX
C. BC XY and AY
D. BC YZ and AX
11. BOY DEA. YB = 4x+8, AD = 60, AE = 80, AND DE = 40. Find x.
12. EXC STP. C = 3x+12, T = 81, and X = 12x β 15 . Find m C . 13. MNODEF. Calculate the value for x, y, and z. πβ π = 50 x = ________ πβ π = 60 πβ π = 70 y = ________ πβ π· = (2π₯ β 20)
πβ πΈ = (1
2π¦ + 10) z = ________
πβ πΉ = (10 + π§) 14. The figure shows . ABCEDC. C is the midpoint of BD and AE. What
reason would you use to prove the triangles are congruent, if any?
If the triangles are congruent, complete the statement: A _____
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A
B C RQ
P P
Q RCB
A P
Q RCB
A
RQ
PA
B C CB
A P
Q R
x
40
8
10
10
2x - 10
2x + 1 x + 580
20
15. If βJKM βRST, how do you know JK RS? A. Definition of a line segment C. SSS Postulate B. CPCTC D. SAS Postulate 16-20 Proving Triangles are Congruent (SSS, SAS, ASA, AAS, HL) For each of the following, give
the reason for triangle congruence.
16. 17. 18.
19. 20.
21& 22. Solve for x 21 22.