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Name: _____________________________________ Period: __________ Geometry Unit 4: Triangles Homework
Show all of your work on a separate sheet of paper. No work = no credit!
Section 4.1: Triangle and Congruency Basics
Find m∠1.
1. 2. 3.
Find the value of the variables and the measures of the angles.
4.
5.
6.
Can you conclude the triangles are congruent? Justify your answer.
7. GHJ and IHJ
8. QRS and TVS
9. FGH and JKH
10. ABC and FGH
Find the values of the variables.
11.
12.
13.
∆CAT ≅ ∆JSD. List each of the following.
14. Three pairs of congruent sides
15. Three pairs of congruent angles
Complete the following proofs.
16. Given: is the angle bisector of ∠ABC. is
the perpendicular bisector of .
Prove: ∆ADB ≅ ∆CDB.
17. Given: and bisect each other. ≅
and ∠A ≅ ∠D.
Prove: ∆ACB ≅ ∆DCE.
Section 4.2: Triangle Congruency by SSS and SAS
Would you use SSS or SAS to prove the triangles congruent? If there is not enough information to prove the triangles
congruent by SSS or SAS, write not enough information.
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2.
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11.
12.
Draw a triangle. Label the vertices A, B, and C.
13. What angle is between and ?
14. What sides include ∠B?
15. What angles include ?
16. What side is included between ∠A and ∠C?
Complete the following proofs.
17. Given: ≅ ≅ .
Prove: ∆ABC ≅ ∆EDC.
18. Given: ≅ .
Prove: ∆WXY ≅ ∆YZX.
19. Given: ≅ ≅ .
Prove: ∆QXR ≅ ∆TXS.
20. Given: is the perpendicular bisector of .
Prove: ∆BAD ≅ ∆BCD.
Section 4.3: Triangle Congruency by ASA and AAS
Name the two triangles that are congruent by ASA.
1. 2. 3.
Would you use ASA or AAS to prove the triangles congruent? If there is not enough information to prove the triangles
congruent by SSS or SAS, write not enough information.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Complete the following proofs.
13. Given: bisects ∠ABC.
Prove: ∆ABD ≅ ∠CBD.
14. Given: ≅ , ∠KJL ≅ ∠MNL.
Prove: ∆JKL ≅ ∆NML.
15. Given: ≅ , ∠PRT ≅ ∠RSP.
Prove: ∆PQT ≅ ∆ RQS.
16. Given: is the angle bisector of ∠ABC and
∠ADC.
Prove: ∆ABD ≅ ∆CBD.
Section 4.4: Congruency with Isosceles, Equilateral, and Right Triangles
Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle.
1. m∠ACB
2. m∠DBC
3. m∠ABC
Find the value of the variables.
4.
5.
6.
7.
8.
9.
10.
11.
12. For what values of x or x and y are the triangles congruent by HL?
13. 14. 15.
What additional information would prove each pair of triangles congruent by the Hypotenuse-Leg Theorem?
16. 17.
18. Complete the following proof.
19. Given: ≅ .
Prove: ∆RUT ≅ ∆RST.
20. Given: bisects , ≅ , ∠ACB and
∠ECD are right angles.
Prove: ∆ACB ≅
∆DCE.
Section 4.5: Corresponding Parts of Congruent Triangles
Explain how you can use SSS, SAS, ASA, or AAS with CPCTC to prove each statement true.
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2.
3.
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9. Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS,
SAS, ASA, AAS, or HL.
10.
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14.
15.
4.5 Homework is continued on the next
page!
Complete each proof.
16. Given: ≅ , ∠B ≅ ∠Y.
Prove: ≅ .
17. Given: ≅ .
Prove: ∠A ≅ ∠E.
18. Given: ≅
Prove: ∆BYA ≅ ∆CXA
19. Given: ≅ , ∠JFH ≅ ∠GHF.
Prove: ≅ .
Section 4.6: Midsegments and Bisectors
Use the diagram to complete the exercise.
1. In ∆MNO, the points C, D, and E are midpoints. CD = 4 cm, CE = 8 cm, and DE = 7cm.
a. Find MO.
b. Find NO.
c. Find MN.
2. In ∆LOB, the points A, R, and T are midpoints. LB = 19 cm, LO = 35 cm, and OB = 29 cm.
a. Find RT.
b. Find AT.
c. Find AR.
Use figure 1 for exercises 3 – 7.
3. How is related to ?
4. Find XV.
5. Find WZ.
6. Find XY.
7. What kind of triangle is ∆WXV?
Use figure 2 for questions 8 – 12.
8. Find the value of x.
9. Find HI.
10. Find JI.
11. If L lies on , then L is _____ from H and J.
12. What kind of triangle is ∆HIJ?
Use figure 3 for questions 13 - 16.
13. Find the value of y.
14. Find PS.
15. Find RS.
16. What kind of triangle is ∆PQS.
Find the value of the variable.
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18.
19.
20.
21.
22.
Use the given measures to identify three pairs of parallel segments in each diagram.
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24.
Section 4.7: Concurrent Lines, Medians, and Altitudes
Is a perpendicular bisector, an angle bisector, an altitude, a median, or none of these?
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2.
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9.
Name each segment.
10. A median in ∆ABC
11. An altitude for ∆ABC
12. A median in ∆AHC
13. An altitude for ∆AHB
14. An altitude for ∆AHG
In ∆ABC, X is the centroid.
15. If CW = 15, find CX and XW.
16. If BX = 8, find BY and XY.
17. If XZ = 3, find AX and AZ.
Name the point of concurrency of the angle bisectors.
18.
19.
20.
21.
22.
23.
Section 4.8: Inequalities in Triangles
Determine the two largest angles in each triangle.
1.
2.
3.
4.
5.
6.
Can a triangle have sides with the given lengths? Explain.
7. 4m, 7m, and 8m
8. 6m, 10m, and 17m
9. 4in, 4in, and 4in
10. 1yd, 9yd, and 9 yd
11. 11m, 12m and 13m
12. 18ft, 20ft, and 40ft
13. 1.2cm, 2.6cm and 5cm
14. 2.5m, 3.5m and 6m
List the sides of each triangle in order from shortest to longest.
15. 16.
17.
List the angles of each triangle in order from largest to smallest.
18. 19. 20.
The lengths of two sides of a triangle are given. Describe the lengths possible for the third side.
21. 4in, 7in 22. 11m, 20m 23. 9cm, 17cm 24. 6km, 8km
Section 4.9: Proving Triangles Similar
In the diagram, ∆PRQ ~ ∆DEF. Find each of the following.
1. The scale factor of ∆PRQ to ∆DEF
2. m∠D
3. m∠R
4. m∠P
5. DE
6. FE
Determine whether the triangles are similar. If so, write a similarity statement and name the postulate or theorem
you used. If not, explain.
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15.
Explain why the triangles are similar. Then find the distance represented by x.
16. 17.
Complete the following proofs.
18. Given:
Prove: ∆RSM ~ ∆STN
19. Given: A bisects , C bisects , B bisects
Prove: ∆JKL ~ ∆CBA
Section 4.10: Similarity in Right Triangles
Identify the following in right ∆QRS.
1. The hypotenuse
2. The segments of the hypotenuse
3. The altitude
4. The segment of the hypotenuse adjacent to leg
Write a similarity statement relating the three triangles in the diagram.
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10.
Solve for the value of the variables in each right triangle.
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Section 4.11: Proportions in Triangles
Solve for x.
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2.
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12.
Use the figure at the right to complete each proportion.
13. a
c f
14. f c
e
15. e
c f
16. a b
e
17. a
b e
18. e f
c