Question

1 a diagram is shown given: (mangle bge + mangle cge=90^{circ}) prove: (mangle agd + mangle cge = 90^{circ}) statement reason (mangle bge + mangle cge = 90) a. (angle bgecongangle agd) b. (mangle bge = mangle agd) c. d. substitution property of equality

Explanation:

Step1: Given information

Given that \(m\angle BGE + m\angle CGE=90^{\circ}\)

Step2: Identify vertical - angles

\(\angle BGE\) and \(\angle AGD\) are vertical angles. Vertical angles are congruent, so \(\angle BGE\cong\angle AGD\) (by the vertical - angles theorem).

Step3: Convert congruence to equality of measures

If two angles are congruent, then their measures are equal. So, if \(\angle BGE\cong\angle AGD\), then \(m\angle BGE = m\angle AGD\)

Step4: Substitute

Substitute \(m\angle BGE\) with \(m\angle AGD\) in the equation \(m\angle BGE + m\angle CGE = 90^{\circ}\). We get \(m\angle AGD + m\angle CGE=90^{\circ}\)

Answer:

a. Given
b. Vertical - angles theorem
c. Definition of congruent angles
d. \(m\angle AGD + m\angle CGE = 90^{\circ}\)