Intermediate Geometry : How to find if acute / obtuse triangles are congruent

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #21 : Acute / Obtuse Triangles

Given:  and .

True or false: It follows from the given information that .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Examine the diagram below.

Untitled

, , and , but . As a result, it is not true that . Therefore, the statement is false.

Example Question #22 : Acute / Obtuse Triangles

Given:  and .

True or false: It follows from the information given that .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The congruence of corresponding angles of two triangles does not alone prove that the triangles are congruent. For example, see the figures below:

Triangles 1

The three angle congruence statements are true, but the sides are not congruent, so the triangles are not congruent. The statement is false.

Example Question #23 : Acute / Obtuse Triangles

Given:  and .

True or false: It follows from the given information that .

Possible Answers:

False

True

Correct answer:

True

Explanation:

As we are establishing whether or not , then , and  correspond respectively to , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. Between  and and  are corresponding sides, their congruence is given. The other two congruences between corresponding sides are given, so the conditions of SSS are satisfied. is indeed true.

Example Question #24 : Acute / Obtuse Triangles

Given:  and .

True or false: It follows from the given information that .

Possible Answers:

True

False

Correct answer:

False

Explanation:

As we are establishing whether or not , then , and  correspond respectively to , and .

By the Side-Side-Side Congruence Postulate (SSS), if all three pairs of corresponding sides of two triangles are congruent, then the triangles themselves are congruent. However, if we restate the first side congruence as

and examine it with the other two:

We see that while we can invoke SSS, the points correspond to , respectively. The triangle congruence that follows is therefore .

The answer is therefore false.

Example Question #25 : Acute / Obtuse Triangles

Given:  and .

True or false: It follows from the given information that .

Possible Answers:

True

False

Correct answer:

True

Explanation:

As we are establishing whether or not , then , and  correspond respectively to , and .

By the Side-Angle-Side Congruence Postulate (SAS), if two pairs of corresponding sides and the included angle of one triangle are congruent to the corresponding parts of a second, the triangles are congruent. and , indicating congruence between corresponding sides, and , indicating congruence between corresponding included angles. This satisfies the conditions of SAS, so is true.

Example Question #1 : How To Find If Acute / Obtuse Triangles Are Congruent

Hinge

Refer to the above two triangles. By what statement does it follow that  ?

Possible Answers:

The Converse of the Isosceles Triangle Theorem

The Angle-Angle-Side Theorem

The Hinge Theorem

The Angle-Side-Angle Postulate

The Converse of the Pythagorean Theorem

Correct answer:

The Angle-Angle-Side Theorem

Explanation:

We are given that two angles of  -   and -  and a nonincluded side  are congruent to their corresponding parts, , and  of  . It follows from the Angle-Angle-Side Theorem that .

Example Question #2 : How To Find If Acute / Obtuse Triangles Are Congruent

Hinge

Refer to the above two triangles. By what statement does it follow that  ?

Possible Answers:

The Hinge Theorem

The Angle-Angle Postulate

The Isosceles Triangle Theorem

The Triangle Midsegment Theorem

The Side-Angle-Side Postulate

Correct answer:

The Side-Angle-Side Postulate

Explanation:

We are given that two sides of  - sides  and  - and their included angle  are congruent to their corresponding parts, sides  and  and  of . It follows from the Side-Angle-Side Postulate that .

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