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4.7 – Equilateral Triangles

4.7 – Equilateral Triangles. Geometry Ms. Rinaldi. Remember that a triangle is equilateral if it has all congruent sides. Equilateral Triangles. Corollary to the Base Angles Theorem. If a triangle is equilateral, then it is equiangular. Corollary to the Converse of Base Angles Theorem.

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4.7 – Equilateral Triangles

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  1. 4.7 – Equilateral Triangles Geometry Ms. Rinaldi

  2. Remember that a triangle is equilateral if it has all congruent sides. Equilateral Triangles

  3. Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular. Corollary to the Converse of Base Angles Theorem If a triangle is equiangular, then it is equilateral.

  4. EXAMPLE 1 Find measures in an Equilateral Triangle Find the measures of angles P, Q, and R. P R SOLUTION Q The diagram shows that the triangle is equilateral. Therefore, it is also equiangular. Since there is 180 degrees in a triangle, we need to split that into 3 equal parts. 180/3 = 60º each.

  5. EXAMPLE 2 Angle Measure in an Equilateral Triangle In Example 1, you found that the angle measures of the equilateral triangle were 60º each. Is it possible for an equilateral triangle to have different angle measures? SOLUTION No. There must be 180º inside a triangle, and the only way to divide that into three equal angles is if each angle is 60º

  6. Find STin the triangle at the right. ThusST = 5 ANSWER STUis equiangular, therefore its is equilateral. Find Measures in Equilateral Triangles EXAMPLE 3 SOLUTION

  7. ALGEBRA Find the values of x and yin the diagram. STEP 1 Find the value of y. Because KLNis equiangular, it is also equilateral and KN KL. Therefore, y = 4. STEP 2 Find the value of x. Because LNM LMN, LN LMand LMNis isosceles. You also know that LN = 4 because KLNis equilateral. EXAMPLE 4 Use isosceles and equilateral triangles SOLUTION (Continued on next slide)

  8. EXAMPLE 4 Use isosceles and equilateral triangles (cont.) LN = LM Definition of congruent segments 4 = x + 1 Substitute 4 for LNand x + 1 for LM. 3 = x Subtract 1 from each side.

  9. Find the values of x and yin the diagram. Use Isosceles and Equilateral Triangles EXAMPLE 5 SOLUTION Since the triangle to the right is equilateral, it is also equiangular. Therefore x and the other angles are 60º. We now know that part of the right angle is 60º. The other part, in the triangle at the left, must be 90 – 60 = 30º. Since that triangle is isosceles, the base angles are congruent. Find y by combining those two angles and subtracting them from 180º. 30+30 = 60 180 – 60 = 120º = y

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