# HiSET: Math : Prove relationships in geometric figures

## Example Questions

### Example Question #1 : Congruence And Similarity Criteria For Triangles

Given:  and  such that and

Does sufficient information exist to prove that , and if so, by what postulate or theorem?

Insufficient information exists to prove that .

can be proved by the AA Similarity Postulate.

can be prove by the SSS Similarity Theorem.

can be proved by the SAS Similarity Theorem.

can be prove by the Isosceles Triangle Theorem.

can be proved by the AA Similarity Postulate.

Explanation:

We are given that, between the triangles, two pairs of corresponding angles are congruent. By the AA Similarity Postulate, this is enough to prove that .

### Example Question #1 : Measurement And Geometry

Given:  and  such that .

Does sufficient information exist to prove that , and if so, by what postulate or theorem?

can be proved by the AA Similarity Postulate.

can be prove by the Isosceles Triangle Theorem.

Insufficient information exists to prove that .

can be proved by the SAS Similarity Theorem.

can be proved by the SAS Inequality Theorem (Hinge Theorem).

Insufficient information exists to prove that .

Explanation:

We are given that, between the triangles, two pairs of corresponding sides are proportional. Without knowing anything else, the proportionality of two pairs of sides is insufficient to prove that the triangles are similar.

### Example Question #2 : Congruence And Similarity Criteria For Triangles

Given:  and  such that and .

Does sufficient information exist to prove that , and if so, by what postulate or theorem?

can be proved by the SAS Similarity Theorem.

Insufficient information exists to prove that .

can be proved by the AA Similarity Postulate.

can be prove by the SSS Similarity Theorem.

can be prove by the Isosceles Triangle Theorem.

can be proved by the SAS Similarity Theorem.

Explanation:

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. The angles that are congruent are the included angles of their respective sides. By the SAS Similarity Postulate, this is enough to prove that .

### Example Question #1 : Congruence And Similarity Criteria For Triangles

with scale factor 5:4, with the larger triangle.

Complete the sentence: the area of is _______% greater than that of .

(Select the closest whole percent)

Explanation:

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of larger triangle  equal to of that of smaller triangle —or, equivalently, greater.

### Example Question #1 : Measurement And Geometry

Given:  and  such that and .

Does sufficient information exist to prove that , and if so, by what postulate or theorem?

can be proved by the SAS Similarity Theorem.

can be proved by the AA Similarity Postulate.

Insufficient information exists to prove that .

can be prove by the Isosceles Triangle Theorem.

can be prove by the SSS Similarity Theorem.

Insufficient information exists to prove that .

Explanation:

We are given that, between the triangles, two pairs of corresponding sides are proportional, and that a pair of corresponding angles are congruent. If the angles were the included angles of the triangles, then the SAS Similarity Theorem could be applied to prove that ; however, the two congruent angles are nonincluded, and there is no "SSA" statement that can be applied to prove similarity. Without further information, it cannot be proved that the triangles are similar.

### Example Question #1 : Prove Relationships In Geometric Figures

with scale factor 4:5, with the smaller triangle.

Complete the sentence: the area of is _______% less than that of .

(Select the closest whole percent).

Explanation:

The ratio of the areas of two similar triangles is equal to the square of their scale factor. The scale factor is equal to , so the ratio of the areas is the square of this, or .

This makes the area of smaller triangle  equal to  of that of larger triangle —or, equivalently,  less.

### Example Question #1 : Congruence And Similarity Criteria For Triangles

with scale factor 5:4, with the larger triangle.

Complete the sentence: the perimeter of is _______% greater than that of .

(Select the closest whole percent).

Explanation:

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of larger triangle  equal to  of that of smaller triangle —or, equivalently, 25% greater.

### Example Question #1 : Measurement And Geometry

with scale factor 4:5, with the smaller triangle.

Complete the sentence: the perimeter of is _______% less than that of .

(Select the closest whole percent)

Explanation:

The ratio of the perimeters of two similar triangles is equal to their scale factor. This is factor is .

This makes the perimeter of the smaller triangle  equal to  of that of larger triangle —or, equivalently,   less.

### Example Question #2 : Congruence And Similarity Criteria For Triangles

A triangle has sides of length 8 and 12; the triangle is scalene and obtuse. Which of the following could be the length of its third side?

Explanation:

A scalene triangle has three sides of different lengths. The triangle is known to have sides of length 8 and 12, so this eliminates 8 and 12 as the correct choices for the length of the third side.

The sum of the lengths of the two smallest sides must exceed the length of the third side. 4 can be eliminated as a the correct choice, since , violating this condition.

This leaves 6 and 10 as possible answers. For a triangle to be obtuse, it must hold that if are its sidelengths, the greatest of the three,

.

If the length of the third side is 10, setting , we see that

,

violating this condition.

If the length of the third side is 6, setting , we see that

,

satisfying this condition.

This makes 6 the correct choice.