37.0k views
2 votes
Answer the questions using the diagram provided.

CE is an angle bisector, CG is an angle bisector.

(a) If m∠FCG = 35, then find m∠FCB.
(b) If m∠DCE = 4x + 15 and m∠ECF = 6x -5, then find m∠DCE.
(c) Verify that the Angle Addition Postulate holds true in this scenario by using the results from part (a) and part (b) using arguments, logic, and/or visual diagrams.

69 easy points if you answer soon!

Answer the questions using the diagram provided. CE is an angle bisector, CG is an-example-1
User Lukewitmer
by
8.2k points

1 Answer

2 votes

Answer:

(a) ∠FCB = 70°

(b) ∠DCE = 55°

(c) ∠DCE +∠ECF +∠FCG +∠GCB = 55° +55° +35° +35° = 180°

Explanation:

Given angle bisectors CE and CG, ∠FCG = 35°, ∠DCE = 4x+15, ∠ECF = 6x-5, you want the measures of angles FCB and DCE.

(a) FCB

Ray CG bisects angle FCB, splitting it into two congruent angles. The measure of one of those (∠FCG) is 35°, so the measure of the two of them together is 2×35° = 70°.

∠FCB = 70°.

(b) DCE

Angles DCE and ECF are created congruent by angle bisector CE. This means ...

∠DCE = ∠ECF

4x +15 = 6x -5

20 = 2x

Then angle DCE is ...

4x +15 = 2(2x) +15 = 2(20) +15 = 55 . . . . . . degrees

∠DCE = 55°

(c) Angle Addition

We have already used the Angle Addition theorem to find the measure of ∠FCB:

∠FCB = ∠FCG +∠GCB = 35° +35° = 70°

We could also use the Angle Addition theorem to find ∠DCE:

∠FCD +∠FCB = 180° . . . . . . . . . the sum of the parts gives the whole

2(∠DCE) +2(∠FCG) = 180° . . . . substitute 2 halves for each angle

∠DCE +∠FCG = 90° . . . . . . . . divide by 2

∠DCE = 90° -∠FCG = 90° -35° = 55° . . . . . matches the above calculation

The sum of all of the parts is the whole:

∠DCE +∠ECF +∠FCG +∠GCB = 55° +55° +35° +35° = 180°

__

Additional comment

You may notice we did not actually find the value of x in part (b). x=10. We didn't need to know this; we only need to know the value of 4x.

The angle addition theorem is similar to the segment addition theorem. It tells you the whole is the sum of the parts. Along with that, we used the fact that a straight angle has a measure of 180°, or a linear pair (FCD, FCB) are supplementary angles (total 180°).

<95141404393>

User Prajul
by
8.9k points

Related questions

asked Jul 20, 2023 56.0k views
Bogl asked Jul 20, 2023
by Bogl
8.0k points
1 answer
1 vote
56.0k views
1 answer
1 vote
85.0k views