# SSAT Upper Level Math : How to find if right triangles are congruent

## Example Questions

### Example Question #1 : How To Find If Right Triangles Are Congruent

Given:

, where  is a right angle; ;

, where  is a right angle and ;

, where  is a right angle and  has perimeter 60;

, where  is a right angle and  has area 120;

, where  is a right triangle and

Which of the following must be a false statement?

All of the statements given in the other responses are possible

Explanation:

has as its leg lengths 10 and 24, so the length of its hypotenuse, , is

Its perimeter is the sum of its sidelengths:

Its area is half the product of the lengths of its legs:

and  have the same perimeter and area, respectively, as ; also, between  and , corresponding angles are congruent. In the absence of other information, none of these three triangles can be eliminated as being congruent to .

However,  and . Therefore, . Since a pair of corresponding sides is noncongruent, it follows that .

### Example Question #1 : How To Find If Right Triangles Are Congruent

Given:  and  with right angles  and .

Which of the following statements alone, along with this given information, would prove that  ?

I)

II)

III)

I or III only

II or III only

III only

I or II only

Any of I, II, or III

Any of I, II, or III

Explanation:

since both are right angles.

Given that two pairs of corresponding angles are congruent and any one side of corresponding sides is congruent, it follows that the triangles are congruent. In the case of Statement I, the included sides are congruent, so by the Angle-Side-Angle Congruence Postulate, . In the case of the other two statements, a pair of nonincluded sides are congruent, so by the Angle-Angle-Side Congruence Theorem, . Therefore, the correct choice is I, II, or III.

### Example Question #3 : How To Find If Right Triangles Are Congruent

, where  is a right angle, , and .

Which of the following is true?

None of the statements given in the other choices is true.

has area 100

has perimeter 40

has area 100

Explanation:

, and corresponding parts of congruent triangles are congruent.

Since  is a right angle, so is  and ; since , it follows that   is an isosceles right triangle; consequently, .

is a 45-45-90 triangle with hypotenuse of length . By the 45-45-90 Triangle Theorem, the length of each leg is equal to that of the hypotenuse divided by ; therefore,

is eliminated as the correct choice.

Also, the perimeter of  is

.

This eliminates the perimeter of  being 40 as the correct choice.

Also,  is eliminated as the correct choice, since the triangle is 45-45-90.

The area of   is half the product of the lengths of its legs:

The correct choice is the statement that  has area 100.