Step: 1

ΔABD and ΔCDB are congruent.

Step: 2

In congruent triangles, the angles opposite to equal sides are equal.

Step: 3

[Alternate interior angles.]

Step: 4

Correct Answer is : ∠ z is congruent to ∠ y .

Step: 1

In triangles SPT and ONM, ∠ S ≅ ∠ M, ∠ P ≅ ∠ O and PT ¯ ≅ ON ¯

[Given.]

Step: 2

Step: 3

If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent.

[AAS theorem.]

Step: 4

ΔSPT ≅ ΔMON

[From step 3.]

Correct Answer is : ΔMON

Step: 1

From the figure ∠ ABC ≅ ∠ QRP, ∠ CAB ≅ ∠ PQR, and BA ¯ ≅ RQ ¯

Step: 2

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

[ASA Postulate.]

Step: 3

So, ΔABC ≅ ΔQRP

Correct Answer is : ΔABC ≅ ΔQRP

Step: 1

ACDB, CEDB are parallelograms in which AB ¯ ≅ CD ¯ , AC ¯ ≅ BD ¯ ≅ CE ¯ and BC ¯ ≅ DE ¯ .

[From the figure.]

Step: 2

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

[SSS postulate.]

Step: 3

So, ΔABC and ΔCDE are congruent to each other.

Correct Answer is : ΔABC ≅ ΔCDE

Step: 1

From the figure,

AB ¯ ≅ PQ ¯

∠ ACD ≅ ∠ PRS

∠ BAC ≅ ∠ QPR

Step: 2

If the two angles and the non included side of one triangle are congruent to the two angles and the non included side of another triangle, then the two triangles are congruent.

[By AAS postulate.]

Step: 3

As the corresponding parts of the triangles are congruent, the triangles are congruent.

Correct Answer is : yes

1. ΔPSR

2. ΔPQS

3.

4.

Step: 1

Given, QT ¯ ≅ RS ¯ .

Step: 2

Step: 3

Consider ΔPSR and ΔPTQ. In ΔPSR, SR = ST + TR and in ΔPTQ, QT = QS + ST ⇒ QT = TR + ST = SR

[Substitute QS = TR.]

Step: 4

Step: 5

So, ΔPSR ≅ ΔPTQ which is true.

[SSS postulate.]

Step: 6

Consider ΔPQS, ΔPRT, QS ¯ ≅ RT ¯ and PS ¯ ≅ PT ¯

Step: 7

So, ΔPQS ≅ ΔPRT which is true.

[SSS postulate.]

Step: 8

[ΔPQS ≅ ΔPRT.]

Correct Answer is : all are correct

Step: 1

Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

Correct Answer is : A pair of triangles have equal sides.

Step: 1

[Given.]

Step: 2

Therefore, by SSS property, ΔABC ≅ ΔFDE.

Correct Answer is : ΔABC ≅ ΔFDE

Step: 1

In ΔDEF, m ∠ E = 90° and m ∠ F = 45°.

Step: 2

Sum of the angles of a triangle is equal to 180°.

Step: 3

Step: 4

[Substitute the measures of angles E and F.]

Step: 5

[Simplify.]

Step: 6

[Subtraction property for equality.]

Step: 7

ΔABC is congruent to ΔDEF.

Step: 8

So, the corresponding angles and sides of ΔABC and ΔDEF are equal.

Step: 9

AB = DE, so, m ∠ F = m ∠ C

[In congruent triangles, the angles opposite to the equal sides are equal. ]

Step: 10

Step: 11

Step: 12

So, the measure of ∠ A is 45°.

Correct Answer is : 45°

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