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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. 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. It looks like a bow tie. For the left triangle, the base is 6. A 23 degree angle is at the upper left corner. and the side opposite the 23 degree angle is, 9. For the triangle at the right, the lower right angle is 23 degrees. The base at the right is, x. On the other side of the 23 degree angle, is 9.

    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. Two right triangles partially connected to each other. The right triangle on the right has the right angle at the upper left with legs at the top and left.  The hypotenuse is, 16. The side opposite the 16 is the right side of the left triangle plus an additional length of, x. The angle opposite the top leg is, 30 degrees. The left triangle has the right angle at the lower right with legs at the right and bottom edges. The hypotenuse is 16. The angle opposite the right side is 30 degrees.

  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.