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Math  /  Geometry

QuestionE.S.T. Geometry EOCME O^{C M} conversions miters in nur =24=24 nous 3ys3 y^{s} MODELING REAL LIFE In Exercises 41 and 42 , use the diagram of the roof truss.
41. BGundefined\overrightarrow{B G} bisects ∠ABC\angle A B C and ∠DEF,m∠ABC=112∘\angle D E F, m \angle A B C=112^{\circ}, and ∠ABC≅∠DEF\angle A B C \cong \angle D E F. Find the measure of each angle. a. ∠DEF\angle D E F b. ∠ABG\angle A B G c. ∠CBG\angle C B G d. ∠DEG\angle D E G

Studdy Solution

STEP 1

1. We are given a roof truss diagram where BGundefined\overrightarrow{BG} bisects ∠ABC\angle ABC and ∠DEF\angle DEF.
2. The measure of ∠ABC\angle ABC is 112∘112^\circ.
3. ∠ABC≅∠DEF\angle ABC \cong \angle DEF, meaning ∠DEF\angle DEF also measures 112∘112^\circ.
4. The bisector BGundefined\overrightarrow{BG} splits ∠ABC\angle ABC into two equal angles.

STEP 2

1. Determine the measure of ∠DEF\angle DEF.
2. Calculate the measure of ∠ABG\angle ABG given that BGundefined\overrightarrow{BG} bisects ∠ABC\angle ABC.
3. Calculate the measure of ∠CBG\angle CBG given that BGundefined\overrightarrow{BG} bisects ∠ABC\angle ABC.
4. Find the measure of ∠DEG\angle DEG given that BGundefined\overrightarrow{BG} bisects ∠DEF\angle DEF.

STEP 3

Since ∠ABC≅∠DEF\angle ABC \cong \angle DEF and ∠ABC=112∘\angle ABC = 112^\circ, it follows that: ∠DEF=112∘ \angle DEF = 112^\circ

STEP 4

Since BGundefined\overrightarrow{BG} bisects ∠ABC\angle ABC, each of the resulting angles ∠ABG\angle ABG and ∠CBG\angle CBG is half of ∠ABC\angle ABC: ∠ABG=12∠ABC \angle ABG = \frac{1}{2} \angle ABC

STEP 5

Calculate ∠ABG\angle ABG: ∠ABG=12×112∘=56∘ \angle ABG = \frac{1}{2} \times 112^\circ = 56^\circ

STEP 6

Since BGundefined\overrightarrow{BG} bisects ∠ABC\angle ABC, it follows that: ∠CBG=12∠ABC \angle CBG = \frac{1}{2} \angle ABC

STEP 7

Calculate ∠CBG\angle CBG: ∠CBG=12×112∘=56∘ \angle CBG = \frac{1}{2} \times 112^\circ = 56^\circ

STEP 8

Since BGundefined\overrightarrow{BG} also bisects ∠DEF\angle DEF, each of the resulting angles ∠DEG\angle DEG and ∠GEF\angle GEF is half of ∠DEF\angle DEF: ∠DEG=12∠DEF \angle DEG = \frac{1}{2} \angle DEF

STEP 9

Calculate ∠DEG\angle DEG: ∠DEG=12×112∘=56∘ \angle DEG = \frac{1}{2} \times 112^\circ = 56^\circ
_FINAL_ANSWERS_: a. ∠DEF=112∘\angle DEF = 112^\circ b. ∠ABG=56∘\angle ABG = 56^\circ c. ∠CBG=56∘\angle CBG = 56^\circ d. ∠DEG=56∘\angle DEG = 56^\circ

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