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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.

  • Math -

    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.

  • Math -

    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.

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