Name_________________________________________________ Date___________________ Class_________
4.7 Challenge: CPCTC
Tell w hich triangles you can show are congruent in order to prove
the statement. W hat postulate or theorem w ould you use?
2. QU ≅ PS
1. ∠ H ≅ ∠ J
3. ∠ A ≅ ∠ D
Use the diagram to w rite a plan for the proof.
4. PRO VE: RZ ≅ YU
5. PRO VE: FH ≅ JH
Use the information given in the diagram to w rite a plan for proving
that ∠ 1 = ∠ 2.
6.
7.
8.
Use the vertices of Δ ABC and Δ DEF to show that ∠ C= ∠ F.
9. A (2, 0), B (2, 2), C (4, l), D (–2, l), E (–5, l), F (–3, –1)
Use the information given in the diagram to w rite a proof.
10. PRO VE: ∠ CJF ≅ ∠ GHE
11. PRO VE: ∠ UTS ≅ ∠ VRQ
C ongruent triangles can help you understand
how humans judge distance. The diagram at the
right show s a triangle formed betw een a person’s
eyes and an object that is straight ahead. The
height h of the triangle is the person’s distance
to the object. The distance betw een the person’s
eyes never changes.
12.
W rite a proof to show that anytime the person’s eyes
focus on an object and form angles congruent to ∠ L
and ∠ R, the distance to the object is the same. ( Hint:
Think of the height as a part of the triangle.)
G IVEN : m ∠L = m ∠ L 1 , m< ∠R = m ∠ R 1 , LR = L 1 R 1 , h and h l are the heights
of the triangle.
PRO VE: h = h 1
Answer Key -- Lesson 4.7 -- Practice Level C
1. Δ HGL ≅ Δ JKM; AAS
2. Δ PQU ≅ Δ VPS; AAS
3. Δ ABC ≅ Δ DEF; ASA
4. Use the ≅ angles in the linear pairs to show ∠ RZS ≅ ∠ UYT. Show
Δ RSZ ≅ Δ UTY by AAS, soRZ ≅ YU because they are corresponding
parts.
Ψ they are vertical. Show Δ FGH ≅ Δ JIH
5. Show ∠ FHG ≅ ∠JHI because
by AAS, so FH ≅ JH because they are corresponding parts.
6. Show ∠ ADE is a right ∠ . Use SAS to show Δ ADE ≅ Δ CDE, so by
corresponding parts, ∠ AED ≅ ∠ CED and AE ≅ . CE Use SAS to show
Δ ABE ≅ Δ CBE. So by corresponding parts, ∠1 ≅ ∠2.
7. Show Δ HKL ≅ Δ HML by ASA, so by corresponding parts, ∠ HKL ≅
∠ HML and HK ≅ HM Use the congruent angles in the linear pairs
to show Δ HMN ≅ Δ HKJ. By vertical angles, ∠ JHK ≅ ∠ NHM. Show
∠ HJK ≅ ∠ HNM by ASA, so by corresponding parts, ∠1 ≅ ∠2 .
8. Use AAS to show Δ ABD ≅ Δ GFD. Then by corresponding parts BD ≅ FD . By vertical angles, ∠ ADB ≅ ∠ EDF and ∠ CDB ≅ ∠ GDF. Show
Δ GFD ≅ Δ EFD by AAS, so by corresponding parts ∠ EFG ≅ ∠ GFD.
Then because they are a ≅ linear pair, ∠ EFG and ∠ GFD are right
angles. Use corresponding parts to show ∠ ABD is a right angle
and linear pair to show ∠ CBD is a right ∠ . Show Δ ABD ≅ Δ CBD by
SAS. Finally, show that by corresponding parts ∠1 ≅ ∠2 .
9. Use the D istance Formula to find the side lengths of the
triangles. Use the SSS C ongruence Postulate to show that
Δ ABC ≅ Δ DEF. Then use the fact that corresponding parts of
congruent triangles are congruent to prove that ∠ C ≅ ∠ F.
10.
Statements
Reasons
1. ∠ C ≅ ∠ G,
1. G iven
∠D ≅ ∠ F,
CD ≅ GF 2. Δ CDH ≅ Δ GFJ
2. ASA C ongruence
3. ∠ FJG ≅ ∠ DHC
3. C orr. parts of ≅ s
4. ∠ FJG and ∠ CJF
4. D efinition of linear
Postulate
are ≅.
are a linear pair,
pair
∠ DHC and ∠ GHD
are a linear pair.
5. ∠ FJG and ∠ CJF
5. Linear Pair Post.
are
supplementary,
∠ DHC and ∠ GHD
are
supplementary.
6. ∠ CJF ≅ ∠ GHE
6. C ongruent
Supplements Thm.
11.
Statements
QT ||SR, Reasons
1. UT|| VR
QU ≅ SV
1. G iven
2. ∠ QUT ≅ ∠ SVR
2. Alt. Exterior
3. ∠ VRS ≅ ∠ UTQ,
3. Alt. Interior Angles
∠RVQ ≅ ∠ TUS
4. Δ QUT ≅ Δ SVR
Angles Thm.
Thm.
4. ASA C ongruence
SU = SV + UV
Post.
5. C orr. parts of ≅ s
are ≅.
6. Angle Addition
Post.
7. QU = SV
7. D ef. of congruent
8. QV = SV + UV
8. Subst. Prop. of
5.
TU ≅ RV
6. QV = QU + UV,
9. QV = SU
10.
QV ≅ SV 11. Δ QRV ≅ Δ STU
12. ∠ UTS ≅ ∠ VRQ
angles
Equality
9. Transitive Prop. of
Equality
10. D ef. of congruent
segments
11. SAS C ogruence
Post.
12. C orr. parts of ≅ s
are ≅.
12.
Statements
1. m ∠ L = m ∠ L 1,
m ∠ R = m ∠ R 1,
LR = L 1 R 1 , h and
h 1 are the heights of the s
.
2. ∠ L ≅ ∠ L 1
∠R ≅ ∠R1
3.
LR ≅ L 1 R 1
4. ΔLRO ≅ Δ L 1 R 1 O 1
5. h = h 1
Reasons
1. G iven
2. D ef. of congruent angles
3. D ef. of congruent
segments
4. ASA C ongruence Post.
5. C orresponding parts of ≅
s are ≅.