4.1.2 Angle Relationships in Triangles (continued) Example 4 Applying the Third Angles Theorem
The Third Angles Theorem states that if there are two triangles such that two angles in one triangle are congruent to two angles in the other triangle, then the third angles in the two triangles must be congruent.
In this example, it is given that ∠ Z and ∠ O are right angles, so they are congruent. It is also given that ∠ Y is congruent to ∠ N . Therefore, two angles in △ XYZ are congruent to two angles in △ MNO . So, by the Third Angles Theorem, ∠ X (the third angle of △ XYZ ) must be congruent to ∠ M (the third angle of △ MNO ). Use the fact that ∠ X ≅ ∠ M to write an equation. Substitute the expressions given in the figure for m ∠ X and m ∠ M into the equation and solve for x 2 . Notice that x 2 is used in each expression, not just x , so solving for x 2 is sufficient. Once the value of x 2 is known, m ∠ X is also known since m ∠ X = x 2 . So, m ∠ X = 40°. Substitute 40° for x 2 in the expression for m ∠ M and simplify to find m ∠ M .
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