mrs. rivas 1. three pairs of congruent sides 2. three pairs of congruent angles

HomeworkMrs. Rivas

4-1 . List each of the following.

1. three pairs of congruent sides

2. three pairs of congruent angles

𝑪𝑨≅ 𝑱𝑺𝑨𝑻 ≅ 𝑺𝑫𝑪𝑻 ≅ 𝑱𝑫

∠𝑪≅∠ 𝑱∠𝑨≅∠𝑺∠𝑻 ≅∠𝑫

HomeworkMrs. Rivas

Algebra Find the values of the variables.

3.

𝟓 𝒙 𝟕𝟒

𝟓 𝒙+𝟕𝟒+𝟑 𝒙+𝟐=𝟏𝟖𝟎𝟖 𝒙+𝟕𝟔=𝟏𝟖𝟎

𝟖 𝒙=𝟏𝟎𝟒𝒙=𝟏𝟑

HomeworkMrs. Rivas

Algebra Find the values of the variables.

4.

𝟐 𝒙=𝟏𝟎𝒙=𝟓

HomeworkMrs. Rivas

Algebra ABCD FGHJ. Find the measures of the given angles or lengths of the given sides.

5.

A B

CD

F G

HJ

𝟑 𝒚=𝒚+𝟓𝟎𝟐 𝒚=𝟓𝟎

𝒚=𝟐𝟓

𝒎𝑩=𝟑 𝒚𝒎𝑩=𝟑(𝟐𝟓)𝒎𝑩=𝟕𝟓

𝒎𝑮=𝒚+𝟓𝟎𝒎𝑮=𝟐𝟓+𝟓𝟎𝒎𝑮=𝟕𝟓

HomeworkMrs. Rivas

Algebra . Find the measures of the given angles or lengths of the given sides.

6.

A B

CD

F G

HJ

𝟐 𝒙+𝟑=𝟑 𝒙+𝟐𝟑=𝒙+𝟐𝟏=𝒙

𝑪𝑫=𝟐 𝒙+𝟑𝑪𝑫=𝟐(𝟏)+𝟑𝑪𝑫=𝟓

𝑯𝑱=𝟑 𝒙+𝟐𝑯𝑱=𝟑 (𝟏)+𝟐𝑯𝑱=𝟓

HomeworkMrs. Rivas

Algebra . Find the measures of the given angles or lengths of the given sides.

7.

A B

CD

F G

HJ

𝟓 𝒛+𝟐𝟎=𝟔 𝒛+𝟏𝟎𝟐𝟎=𝒛+𝟏𝟎𝟏𝟎=𝒛

𝒎𝑪=𝟓 𝒛+𝟐𝟎𝒎𝑪=𝟓 (𝟏𝟎 )+𝟐𝟎𝒎𝑪=𝟕𝟎

𝒎𝑯=𝟓 𝒛+𝟐𝟎𝒎𝑪=𝟓 (𝟏𝟎 )+𝟐𝟎𝒎𝑪=𝟕𝟎

HomeworkMrs. Rivas

Algebra . Find the measures of the given angles or lengths of the given sides.

8.

A B

CD

F G

HJ

𝟓𝒃+𝟒=𝟑𝒃+𝟖𝟐𝒃+𝟒=𝟖

𝟐𝒃=𝟒𝒃=𝟐

𝑨𝑫=𝟓𝒃+𝟒𝑨𝑫=𝟓(𝟐)+𝟒𝑨𝑫=𝟏𝟒

𝑭 𝑱=𝟑𝒃+𝟖𝑭 𝑱=𝟑 (𝟐)+𝟖𝑭 𝑱=𝟏𝟒

HomeworkMrs. Rivas

4-2 Draw . Use the triangle to answer the questions below.

9. What angle is included between and ?

M

G

T

∠𝑴

HomeworkMrs. Rivas

4-2 Draw . Use the triangle to answer the questions below.10. Which sides include ?

M

G

T

𝑻𝑮𝒂𝒏𝒅𝑻𝑴

HomeworkMrs. Rivas

4-2 Draw . Use the triangle to answer the questions below.

11. What angle is included between and ?

M

G

T

∠𝑮

HomeworkMrs. Rivas

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. Explain your answer.

12. 13.

Not enough information; two pairs of corresponding sides are congruent, but the congruent angle is not included.

SAS; two pairs of corresponding sides and their included angle are congruent.

HomeworkMrs. Rivas

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. Explain your answer.

14. 15.

SSS; three pairs of corresponding sides are congruent.

Not enough information; two pairs of corresponding sides are congruent, but the congruent angle is not the included angle.

HomeworkMrs. Rivas

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. Explain your answer.

16. 17.

SSS; three corresponding sides are congruent

SAS; two pairs of corresponding sides and their included right angle are congruent.

HomeworkMrs. Rivas

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. Explain your answer.

18. 19.

Not enough information; one pair of corresponding sides and corresponding angles are congruent, but the other pair of corresponding sides that form the included angle must also be congruent.

SAS; two pairs of corresponding sides and their included vertical angles are congruent.

HomeworkMrs. Rivas

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. Explain your answer.

20.

SSS or SAS; three pairs of corresponding sides are congruent, or, two pairs of corresponding sides and their included vertical angles are congruent.

HomeworkMrs. Rivas

4-3 Name two triangles that are congruent by ASA.

21. 22.

∆ 𝑯𝑰𝑱 ≅ ∆𝑴𝑳𝑲 ∆ 𝑹𝑺𝑻 ≅ ∆𝑿𝒀𝒁

HomeworkMrs. Rivas

23. Developing Proof Complete the proof by filling in the blanks. Given: , and

Prove:

Proof: and are given.

𝐼𝐽 ≅ 𝐼𝐽 by _____.

So, by _____.

𝑹𝒆𝒇𝒍𝒆𝒙𝒊𝒗𝒆 𝑪𝒐𝒏𝒈𝒓𝒖𝒆𝒏𝒕 𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒚

𝑨𝑺𝑨

HomeworkMrs. Rivas

24. Given: ,

Prove:

Proof: and are given.

So, by AAS.

because vertical angles are .

HomeworkMrs. Rivas

(4-4) 1. Complete the proof.

Given: , ,

Prove:

Statements Reasons

a) , a)

b) and are right angles. b) Definition of right angles

c) c)

d) d) Vertical angles are congruent.

e) e)

f) f)

g) g)

Given

All right angles are congruent∠𝑬𝑪𝑫≅∠𝑨𝑪𝑩Given∆𝑪𝑫𝑬 ≅ ∆𝑪𝑩𝑨 ASA

CPCTC

HomeworkMrs. Rivas

(4-5) Algebra Find the values of and .2.

𝟏𝟖𝟎−𝟏𝟏𝟓=𝟔𝟓𝟔𝟓𝟔𝟓

𝒙=𝟔𝟓

𝑹𝒆𝒎𝒐𝒕𝒆𝒂𝒏𝒈𝒍𝒆𝒔𝒚+𝟔𝟓=𝟏𝟏𝟓𝒚=𝟓𝟎

base angles are congruent

HomeworkMrs. Rivas

Algebra Find the values of and .

3. all sides are equal and all angles are

equal

𝟔𝟎𝒙+𝟓=𝟔𝟎 𝒚 −𝟏𝟎=𝟔𝟎𝒙=𝟓𝟓 𝒚=𝟕𝟎𝟔𝟎

𝟔𝟎

HomeworkMrs. Rivas

Algebra Find the values of and .

4.

𝑹𝒆𝒎𝒐𝒕𝒆𝒂𝒏𝒈𝒍𝒆𝒔𝒚+𝟗𝟎=𝟏𝟏𝟎

𝒚=𝟐𝟎

𝒙+𝟏𝟏𝟎=𝟏𝟖𝟎𝒙=𝟕𝟎

HomeworkMrs. Rivas

Algebra Find the values of and .

5.

base angles are congruent

𝟒𝟓+𝟒𝟓+𝒙=𝟏𝟖𝟎𝟗𝟎+𝒙=𝟏𝟖𝟎𝒙=𝟗𝟎

𝒚+𝒚+𝟏𝟐𝟎=𝟏𝟖𝟎𝟐 𝒚+𝟏𝟐𝟎=𝟏𝟖𝟎

𝟐 𝒚=𝟔𝟎𝒚=𝟑𝟎

HomeworkMrs. Rivas

(4-5) Algebra Find the values of and .6.

all sides are equal and all angles are equal𝟒 𝒙=𝟔𝟎

𝒙=𝟏𝟓𝑹𝒆𝒎𝒐𝒕𝒆𝒂𝒏𝒈𝒍𝒆𝒔𝒚=𝟔𝟎+𝟔𝟎𝒚=𝟏𝟐𝟎

𝟔𝟎

𝟔𝟎 𝟔𝟎

HomeworkMrs. Rivas

(4-5) Algebra Find the values of and .7.

base angles are congruent

𝟑𝟕 𝟑𝟕 𝒚=𝟑𝟕

𝟑𝟕

𝟑𝟕+𝟑𝟕+𝒙+𝟑𝟕=𝟏𝟖𝟎𝟏𝟏𝟏+𝒙=𝟏𝟖𝟎

𝒙=𝟔𝟗

HomeworkMrs. Rivas

Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle.

8.

𝟒𝟓

𝟒𝟓+𝒎∠𝑨𝑪𝑩=𝟏𝟖𝟎𝒎∠𝑨𝑪𝑩=𝟏𝟑𝟓

HomeworkMrs. Rivas

Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle.

9.

𝟕𝟎

𝟕𝟎+𝟗𝟎+𝒎∠𝑫𝑩𝑪=𝟏𝟖𝟎𝟏𝟔𝟎+𝒎∠𝑫𝑩𝑪=𝟏𝟖𝟎

𝒎∠𝑫𝑩𝑪=𝟐𝟎

HomeworkMrs. Rivas

Use the properties of isosceles and equilateral triangles to find the measure of the indicated angle.

10.

𝟓𝟓

𝟓𝟓+𝟓𝟓+𝒎∠𝑫𝑪𝑬=𝟏𝟖𝟎𝟏𝟏𝟎+𝒎∠𝑫𝑪𝑬=𝟏𝟖𝟎

𝒎∠𝑫𝑪𝑬=𝟕𝟎𝟕𝟎

𝟕𝟎𝟓𝟓 𝟓𝟓

𝒎∠𝑨𝑩𝑪=𝟓𝟓

HomeworkMrs. Rivas

Algebra Find the values of m and n.

11.

𝟒𝟓

𝒎=𝟒𝟓

(𝒏+𝟒𝟓)+(𝒏+𝟒𝟓)+𝟔𝟎=𝟏𝟖𝟎𝟐𝒏+𝟗𝟎+𝟔𝟎=𝟏𝟖𝟎𝟐𝒏+𝟏𝟓𝟎=𝟏𝟖𝟎

𝟐𝒏=𝟑𝟎𝒏=𝟏𝟓

HomeworkMrs. Rivas

Algebra Find the values of m and n.

12.

𝟔𝟖

𝟔𝟖+𝟔𝟖+?=𝟏𝟖𝟎? 𝟏𝟑𝟔+?=𝟏𝟖𝟎

?=𝟒𝟒𝟒𝟒

𝟒𝟒

𝒎=𝟒𝟒

𝒏=𝟔𝟖

HomeworkMrs. Rivas

Algebra Find the values of m and n.

13.

base angles are congruent

𝒏+𝒏+𝟗𝟎=𝟏𝟖𝟎𝟐𝒏+𝟗𝟎=𝟏𝟖𝟎

𝟐𝒏=𝟗𝟎𝒏=𝟒𝟓

𝟒𝟓

𝟒𝟓𝟒𝟓

𝒎+𝒎+𝟒𝟓=𝟏𝟖𝟎𝟐𝒎+𝟒𝟓=𝟏𝟖𝟎

𝟐𝒎=𝟏𝟑𝟓𝒎=𝟔𝟕 .𝟓

HomeworkMrs. Rivas

(4-6) Algebra For what values of x or x and y are the triangles congruent by HL?

14.

𝟒 𝒙+𝟏=𝟏𝟑𝟒 𝒙=𝟏𝟐𝒙=𝟑

𝟐 𝒙+𝒚=𝟖 𝒙−𝟐 𝒚𝟐(𝟑)+𝒚=𝟖(𝟑)−𝟐 𝒚𝟔+𝒚=𝟐𝟒−𝟐𝒚𝟔+𝟑 𝒚=𝟐𝟒

𝟑 𝒚=𝟏𝟖𝒚=𝟔

HomeworkMrs. Rivas

(4-6) Algebra For what values of or and are the triangles congruent by ?

15.

𝟐 𝒙+𝟏=𝒙+𝟑𝒙+𝟏=𝟑𝒙=𝟐

𝒙+𝒚=𝟑𝒙 −𝟑 𝒚(𝟐 )+𝒚=𝟑 (𝟐 ) −𝟑 𝒚𝟐+𝒚=𝟔−𝟑 𝒚𝟐+𝟒 𝒚=𝟔𝟒 𝒚=𝟒𝒚=𝟏

HomeworkMrs. Rivas

(4-6) Algebra For what values of or and are the triangles congruent by ?

16.

𝒚+𝒙=𝒙+𝟕𝒚=𝟕

𝒚+𝟓=𝟐 𝒚 −𝒙𝟕+𝟓=𝟐(𝟕)−𝒙𝟏𝟐=𝟏𝟒− 𝒙𝒙=𝟐

HomeworkMrs. Rivas

17. Developing Proof Complete the paragraph proof.

Given:

Prove:

Proof: It is given that . So, _____ and _______are _______ angles

because perpendicular lines form ______ angles. _______ by the

Reflexive Property of Congruence. It is given that _______. So,

by _______.

∠𝑹𝑻𝑺 ∠𝑹𝑻𝑼 𝒓𝒊𝒈𝒉𝒕

𝒓𝒊𝒈𝒉𝒕 𝑹𝑻

𝑹𝑼𝑯𝑳

HomeworkMrs. Rivas

(4-7) Separate and redraw the indicated triangles. Identify any common angles or sides.

16. ∆𝐵𝑌𝐴∧∆𝐶𝑋𝐴

∠𝑨

HomeworkMrs. Rivas

(4-7) Separate and redraw the indicated triangles. Identify any common angles or sides.

17. ∆𝐺𝐸𝐻∧∆𝐹𝐸𝐻

𝑬𝑯

HomeworkMrs. Rivas

(4-7) Separate and redraw the indicated triangles. Identify any common angles or sides.

18. ∆𝑀𝑃𝑁∧∆𝑀𝑂𝑄

𝑵𝒐𝒏𝒆