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8-44.

If the triangles in each diagram below are congruent, state which triangle congruence condition can be used. If the triangles are not congruent or if there is not enough information, state, “Cannot be determined”. Then calculate the value of in each diagram, if possible.

Read the Math Notes box in Lesson 7.1.7 on the triangle congruence conjectures.
Are the two triangles in each part congruent?

  1. A parallelogram. The top and bottom sides, each labeled, 7. There is a diagonal from the upper left to lower right corners. The upper right angle is X. The upper left angle above the diagonal is 73 degrees. The upper left angle below the diagonal is 28 degrees. The bottom right angle below the diagonal is 73 degrees.

    To prove the triangles congruent notice both triangles have a side that is .
    Also, use alternate interior angles with the angle.
    Finally both triangles share the diagonal so based on the reflexive property these sides are congruent.
    Finally the triangles are congruent by congruence.
    So using the triangle sum theorem, .

  1. Two line segments intersect forming two triangles when a line segment goes across the ends at the left and right sides. For the left triangle, the left side is 6, bottom right side is 9, & top left angle is 23 degrees, For the right triangle, the right side is x, the bottom left side is 9, the lower right angle is 23 degrees.\

    Cannot be determined, because the triangles cannot be found to be congruent.
    If the angle was between the and the on both triangles
    then the triangles would be congruent by . Since they are not then the triangles are not congruent.

  1. 2 right triangles, each with hypotenuse of 16 & angle opposite short leg, 30 degrees, arranged so left triangle has right angle at bottom right, triangle on right, has right angle at top left, short vertical leg of left, is part of the long vertical leg of right, with the long vertical leg extending below the horizontal leg of left, a distance of x.

  1. Two right triangles intersect. The triangle at the right has the 90 degree angle at the lower right corner with the right side, 5, and a bottom side, 2, plus 7. The triangle at the left has the 90 degree angle at the lower left corner with the left side, x, and a bottom side, 2, plus 7. Both with sides, labeled, x, and 5, each have 2 tick marks. Both triangles intersect on the length of 7.