# 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 #491 : Geometry

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

Any of I, II, or III

I or II only

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 #51 : Right Triangles , where is a right angle, , and .

Which of the following is true?  has perimeter 40

None of the statements given in the other choices is true.  has area 100 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.

### All SSAT Upper Level Math Resources 