### All Intermediate Geometry Resources

## Example Questions

### Example Question #28 : Triangles

Refer to the above diagram. .

True or false: From the information given, it follows that .

**Possible Answers:**

True

False

**Correct answer:**

True

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar.

and are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, .

and are alternating interior angles formed by two parallel lines and cut by a transversal . As a consequence, .

The conditions of the Angle-Angle Similarity Postulate are satisfied, and it holds that .

### Example Question #21 : Acute / Obtuse Isosceles Triangles

Refer to the above diagram. .

True or false: From the information given, it follows that .

**Possible Answers:**

True

False

**Correct answer:**

False

The given information is actually inconclusive.

By the Angle-Angle Similarity Postulate, if two pairs of corresponding angles of a triangle are congruent, the triangles themselves are similar. Therefore, we seek to prove two of the following three angle congruence statements:

and are a pair of vertical angles, having the same vertex and having sides opposite each other. As such, .

, but this is not one of the statements we need to prove. Also, without further information - for example, whether and are parallel, which is *not* given to us - we have no way to prove either of the other two necessary statements.

The correct response is "false".

### Example Question #30 : Triangles

and are both isosceles triangles;

True or false: from the given information, it follows that .

**Possible Answers:**

False

True

**Correct answer:**

False

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

is an isosceles triangle, so it must have two congruent angles. has measure , so either has this measure, has this measure, or . If we examine the second case, it immediately follows that . One condition of the similarity of triangles is that all pairs of corresponding angles be congruent; since there is at least one case that violates this condition, it does not necessarily follow that . This makes the correct response "false".

### Example Question #31 : Triangles

is an equilateral triangle; is an equiangular triangle.

True or false: From the given information, it follows that .

**Possible Answers:**

False

True

**Correct answer:**

True

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

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is ). It follows that all angles of both triangles measure .

Specifically, and , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that , making the correct answer "true".

### Example Question #32 : Triangles

Given: and such that

Which statement(s) *must* be true?

(a)

(b)

**Possible Answers:**

(a) but not (b)

(a) and (b)

Neither (a) nor (b)

(b) but not (a)

**Correct answer:**

(a) but not (b)

The sum of the measures of the interior angles of a triangle is , so

Also,

,

so

By similar reasoning, it holds that

Since , by substitution,

Therefore,

,

or

This, along with the statement that , sets up the conditions of the Angle-Angle Similarity Postulate - if two angles of one triangle are congruent to the two corresponding angles of another triangle, the two triangles are similar. It follows that

.

However, congruence cannot be proved, since at least one side congruence is needed to prove this. This is not given in the problem.

Therefore, statement (a) must hold, but not necessarily statement (b).

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