Question

H146.25° F D 67.5° E G In the diagram, ZGEF and ZHFD are exterior angles of ADEF; mZGEF = 67.5°; and mZHFD = 146.250 Prove that ADEF is isosceles by choosing from the options provided in each row. Statement Reason ADEF: exterior angles GEF and HFD. Given information OR Exterior angle theorem mZGEF = 67.5°: and m_HFD = 146.25 ZDEF and ZGEF are supplementary angles Substitution OR Linear pair theorem OR Exterior angle theorem MZDEF+mZGEF = 180° Def of supplementary angles OR Substitution OR Exterior angle theorem m_DEF+67.5° = 180° Def of supplementary angles OR Substitution OR Angle addition postulate mZDEF = 112.5 Subtraction property OR Substitution mZEDF+mZDEF=mZHED Triangle sum theorem OR Angle addition postulate OR Exterior angle theorem mZEDF +112.5° = 146.25 Substitution OR Angle addition postulate OR Addition property mZEDF = 33.75 Substitution OR Angle addition postulate OR Subtraction property m_EDF+mZDEF +mZDFE = 180° Angle addition postulate OR Triangle sum theorem 33.75° +112.5° +mZDFB = 180° Angle addition postulate OR Substitution mZDFE = 33.75 Subtraction property OR Substitution OR Angle addition postulate DEFE Def of an isosceles triangle OR Converse of an isosceles triangle A DEF is isosceles Def of an isosceles triangle OR Converse of an isosceles triangle

Answer 1

Triangle ADEF is isosceles.

The given angles ZGEF and ZHFD are exterior angles of triangle ADEF, with measures 67.5° and 146.25°, respectively. Using the Exterior Angle Theorem and properties of supplementary angles, we establish that ADEF is isosceles, where angles ZEDF and ZDEF are congruent.

In the given diagram, we are asked to prove that triangle ADEF is isosceles. We are provided with the information that angles ZGEF and ZHFD are exterior angles of triangle ADEF, where angle ZGEF measures 67.5° and angle ZHFD measures 146.25°. To prove that ADEF is isosceles, we can utilize the Exterior Angle Theorem and the properties of supplementary angles.

Firstly, we establish that angles ZGEF and ZHFD are exterior angles of triangle ADEF based on the given information. Then, we use the Exterior Angle Theorem to state that the sum of an exterior angle and its corresponding interior angle is 180°. Applying this theorem, we set up an equation involving the measure of angle ZGEF and angle ZDEF, concluding that ZDEF and ZGEF are supplementary angles.

Next, we utilize the property of supplementary angles to express the sum of angles ZDEF and ZGEF as 180°. By solving the equation, we find that the measure of angle ZDEF is 112.5°. Now, we apply the Triangle Sum Theorem, stating that the sum of interior angles in a triangle is 180°. We set up an equation involving angles ZEDF and ZDEF, finding that the measure of angle ZEDF is 33.75°.

Finally, we invoke the definition of an isosceles triangle, stating that a triangle is isosceles if it has at least two congruent sides. In this case, angles ZEDF and ZDEF are congruent, leading to the conclusion that ADEF is an isosceles triangle.