Question

given: (mangle bge + mangle cge=90^{circ}) prove: (mangle agd + mangle cge = 90^{circ}) complete the two - column proof below. statement reason 1 (mangle bge + mangle cge = 90^{circ}) 1 given 2 (angle bgecongangle agd) 2 3 (mangle bge = mangle agd) 3 4 4 substitution property of equality

Explanation:

Step1: Identify vertical - angle relationship

$\angle BGE$ and $\angle AGD$ are vertical angles.

Step2: Recall vertical - angle property

Vertical angles are congruent, so $\angle BGE\cong\angle AGD$.

Step3: Use congruence to get equal measures

If two angles are congruent, their measures are equal, so $m\angle BGE = m\angle AGD$.

Step4: Substitute

Substitute $m\angle BGE$ with $m\angle AGD$ in $m\angle BGE + m\angle CGE=90^{\circ}$, getting $m\angle AGD + m\angle CGE = 90^{\circ}$.

Answer:

Statement	Reason
2. $\angle BGE\cong\angle AGD$	2. Vertical angles are congruent
3. $m\angle BGE = m\angle AGD$	3. Congruent angles have equal measures
4. $m\angle AGD + m\angle CGE = 90^{\circ}$	4. Substitution property of equality