4.2.5 Isosceles and Equilateral Triangles (continued) Example 2 Finding the Measure of an Angle
In this example the measure of only one angle is given. m ∠ R = 28° Additionally, from the figure you know that two sides of △ PQR are also congruent. The Isosceles Triangle Theorem can be applied to any triangle with two congruent sides. So, by the Isosceles Triangle Theorem, the angles opposite of the each congruent side must also be congruent. Therefore, if m ∠ P = x °, then m ∠ Q = x ° as well. Now use the Triangle Sum Theorem to write an equation where the sum of the angles is equal to 180°. Then, solve this equation for x . Since m ∠ Q = x °, the value of x is the answer. The given triangle, △ ABC , is an isosceles triangle since two of its sides, AB and AC are given to be congruent. So, the Isosceles Triangle Theorem can be applied. By the Isosceles Triangle Theorem, ∠ B is congruent to ∠ C since ∠ B and ∠ C are the angles opposite of the congruent sides. Then by the definition of congruence, the measure of the two angles must be equal. So, write an equation using the expression given for m ∠ B , ( x + 44)°, and the expression given for m ∠ C , 3 x °. Solve this equation for x and then substitute the value of x back into the expression for m ∠ C to find that angle’s measure.
Example 3 Using Properties of Equilateral Triangles
An equilateral triangle has three congruent sides and an equiangular triangle has three congruent angles. The Equilateral Triangle Theorem and the Equiangular Triangle Theorem basically state that being one of those two types of triangles implies the other. Specifically, by the Equilateral Triangle Theorem, if it is known that a triangle is equilateral, then it can be assumed that the triangle is also equiangular. And by the Equiangular Triangle Theorem, if it is known that a triangle is equiangular, then it can be assumed that the triangle is also equilateral. So, all equilateral triangles are equiangular and all equiangular triangles are equilateral.
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