Question

geometry - opt 1 - topic 1 & topic 2 combined performance task
alice wants to prove the following:
given abcde with diagonals db and ec and intersection point g.
prove m∠egb = m∠dgc and m∠dge = m∠bgc.
complete the proof
statement reason

raheem wants to prove that the sum of the angle measures of abcde is equal to 540. write an if - then statement raheem can use to prove his conjecture.
using the fact that the sum of the angle measures in a quadrilateral is equal to 360, complete raheems proof.

Explanation:

Step1: Recall vertical - angle property

Vertical angles are equal. Since ∠EGB and ∠DGC are vertical angles (formed by the intersection of lines DB and EC), we have m∠EGB=m∠DGC.

Step2: Recall angle - addition postulate

Let's assume we can show the relationship between angles based on the fact that if we consider the angles around point G. But for the second part m∠DGE = m∠BGC, we can also use the vertical - angle property as ∠DGE and ∠BGC are vertical angles formed by the intersection of DB and EC.
For Raheem's problem:

Step3: Divide the pentagon into triangles or use angle - sum formula

The sum of the interior angles of an n - sided polygon is given by (n - 2)×180°. For a pentagon, n = 5, so (5 - 2)×180°=540°. An if - then statement could be: If ABCDE is a pentagon, then the sum of the angle measures of ABCDE is 540°.
We can also divide the pentagon ABCDE into three non - overlapping triangles. The sum of the angle measures of a triangle is 180°. So the sum of the angle measures of three triangles is 3×180° = 540°.

Answer:

For the first part:

Statement	Reason
m∠DGE = m∠BGC	Vertical angles are equal

For Raheem's if - then statement: If ABCDE is a pentagon, then the sum of the angle measures of ABCDE is 540°.
For Raheem's proof: Divide pentagon ABCDE into three non - overlapping triangles. Since the sum of the angle measures of a triangle is 180°, the sum of the angle measures of the pentagon is 3×180°=540°.