Prove: If the hypotenuse and an acute angle of one triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

Hmmm. In a right triangle, if you know one acute angle, you automratically know the other (90-givenAcuteAngle).

You have ASA then the same, which means congruency.

Recall that the criteria for our congruence postulates have called for three pairs of congruent parts between triangles. The HL Theorem essentially just calls for congruence between two parts: the hypotenuse and a leg. Let's look at an illustration that shows the correct way to use the Hypotenuse-Leg Theorem.

say we have a diagram of a triangle with point at the top A, left bottom point b and right bottom C. we have a side ways triangle with point to the right d, point at top F and point at bottom E. we have congruent hypotenuses (AB?DE) and congruent legs (CA?FD).

We are ready to begin practicing with the HL Theorem. Let's go through the following exercises to get a feel for how to use this helpful theorem.
same triangle set up at above but letters are different. 1ST TRIANGLE POINT AT TOP Q, S BOTTOM LEFT AND R BOTTOM RIGHT, OTHER TRIANGLE POINT OF TOP OF TRIANGLE T, TOP POINT V AND BOTTOM POINT U.

Exercise 1
What additional information do we need in order to prove that the triangles below are congruent by the Hypotenuse-Leg Theorem?

Answer:

Notice that both triangles are right triangles because they both have one right angle in them. Therefore, if we can prove that the hypotenuses of the triangles and one leg of each triangle are congruent, we will be able to apply the HL Theorem.

Looking at the diagram, we notice that segments SQ and VT are congruent. Recall that the side of a right triangle that does not form any part of the right angle is called the hypotenuse. So, the diagram shows that we have congruent hypotenuses.

To prove the statement, we need to use the Side-Angle-Side (SAS) congruence criterion for triangles. According to this criterion, if two triangles have two sides and the included angle of one triangle congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

Given:
Triangle ABC and Triangle DEF

Conditions:
1. Hypotenuse AC is congruent to hypotenuse DF.
2. Angle A is congruent to angle D.

To prove:
Triangle ABC is congruent to Triangle DEF.

Proof:
1. Draw triangles ABC and DEF, with right angles at B and E.
2. Since angle A is congruent to angle D (given), we can mark angle D congruent to angle A.
3. Since angle A and angle D are congruent, they are both right angles.
4. Therefore, ABC and DEF are both right triangles.
5. By definition, a right triangle is a triangle with a right angle.
6. Since both triangles have a right angle, they both satisfy the definition of a right triangle.
7. Now, let's consider the hypotenuses. AC is congruent to DF (given).
8. By definition, the hypotenuse is the side opposite the right angle in a right triangle.
9. Since both hypotenuses are congruent, they are the longest sides in their respective triangles.
10. Also, since both triangles are right triangles, the hypotenuses are opposite their corresponding right angles.
11. Using the SAS congruence criterion, we have two sides (AC and DF) and the included angle (angle A or angle D) congruent in both triangles.
12. Therefore, by the SAS congruence criterion, triangle ABC is congruent to triangle DEF.
13. This completes the proof.

Hence, if the hypotenuse and an acute angle of one triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

To prove the statement, we can use the Side-Angle-Side (SAS) congruence criterion. According to this criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Let's denote the two triangles as ∆ABC and ∆DEF, where the right angles are at points C and F, respectively. Given that the hypotenuse and an acute angle of ∆ABC are congruent respectively to the hypotenuse and an acute angle of ∆DEF, we can state:

Hypotenuse: AC ≅ DE
Acute angle: ∠BAC ≅ ∠EDF

Now, we need to show that the third side of each triangle is also congruent. In a right triangle, the third side is uniquely determined once we know the length of the hypotenuse and one acute angle.

Since ∠BAC ≅ ∠EDF, we have a correspondence between the sides of the triangles.

Using this correspondence:
Side BC corresponds to side EF

Therefore, BC ≅ EF

By the SAS congruence criterion, we have proven that if the hypotenuse and an acute angle of one triangle are congruent respectively to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.