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Basic Geometry : How to find if right triangles are congruent

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Example Question #301 : Right Triangles

Are these right triangles congruent?

Congruent right triangles

Possible Answers:

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Yes - by the angle-angle-side theorem

No - the angles are different

Cannot be determined - we need at least one pair of angles, or all three sides

No - at least one pair of corresponding sides is not congruent

Correct answer:

Yes - all three pairs of sides must be congruent by Pythagorean Theorem

Explanation:

Right now we can't directly compare these triangles because we do not know all three side lengths. However, we can use Pythagorean Theorem to determine both missing sides. The left triangle is missing the hypotenuse:

$\small 48^2 + 55^2 = c^2$

$\small 2,304 + 3,025 = c^2$

$\small 5,329 = c$

$\small c = \sqrt{5,329}= 73$

The right triangle is missing one of the legs:

$\small x^2 + 48^2 = 78^2$

$\small x^2 + 2,304 = 5,329$ subtract 2,304 from both sides

$\small x^2 = 3,025$

$\small x = \sqrt{3,025} = 55$

This means that the two triangles both have side lengths 48, 55, 73, so they must be congruent.

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Example Question #11 : How To Find If Right Triangles Are Congruent

The hypotenuse and acute angle are given for several triangles. Which if any are congruent? Triangle A- Hypotenuse=15; acute angle=56 degrees. Triangle B- Hypotenuse=18; acute angle=56 degrees. Triangle C-Hypotenuse=18; acute angle= 45 degrees.

Possible Answers:

B & C

A & C

None of these

All three.

A & B

Correct answer:

None of these

Explanation:

The correct answer is none of these. There are several pairs of angles and sides or sides and angles that must be the same in order for two triangles to be congruent.

In our case, we need the acute angle and the hypotenuse to both be equal. No two triangles above have this relationship and therefore no two are congruent.

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Example Question #303 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB } \cong \overline{DE}$

$\overline{AC} \cong \overline{DF}$

True or false: From the above information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

If we seek to prove that $\bigtriangleup ABC \cong \bigtriangleup DEF$ , then , , and correspond to , , and , respectively.

By the Hypotenuse-Leg Theorem (HL), if the hypotenuse and one leg of a triangle are congruent to those of another, the triangles are congruent.

$\angle B$ and $\angle E$ are both right angles, so $\bigtriangleup ABC$ and $\bigtriangleup DEF$ are both right triangles. $\overline{AB }$ and $\overline{DE}$ are congruent corresponding sides, and moreover, since, each includes the right-angle vertex as an endpoint, they are congruent corresponding legs. $\overline{AC}$ and $\overline{DF}$ are opposite the right angles, making them congruent corresponding hypotenuses.

The conditions of HL are satisfied, so $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

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Example Question #304 : Right Triangles

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ and $\angle E$ are both right angles.

$\overline{AB} \cong \overline{BC}$

$\overline{DE} \cong \overline{EF}$

True or false: From the given information, it follows that $\bigtriangleup ABC \cong \bigtriangleup DEF$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

The congruence of $\bigtriangleup ABC$ and $\bigtriangleup DEF$ cannot be proved from the given information alone. Examine the two triangles below:

Triangles 2

$\overline{AB} \cong \overline{BC}$ , $\overline{DE} \cong \overline{EF}$ , and $\angle B$ and $\angle E$ are both right angles, so the conditions of the problem are met; however, since the sides are not congruent between triangles - for example, $\overline{AB} \ncong \overline{DE}$ - the triangles are not congruent either.

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Basic Geometry : How to find if right triangles are congruent

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #301 : Right Triangles

Example Question #11 : How To Find If Right Triangles Are Congruent

Example Question #303 : Right Triangles

Example Question #304 : Right Triangles

All Basic Geometry Resources

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