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QuestionMathematics
"What is m∠DEF?
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There is a triangle DEF in which side DE is congruent to side EF and G is the midpoint of side DF. Segment EG intersects side DF at point G. The measure of angle DEG is (3y+ 4) degrees and the measure of angle FEG is (5y−10) degrees."
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Answer
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Step 1:Let's solve this step by step:
Step 2:: Analyze the given information
- Triangle DEF has sides DE = EF (isosceles triangle) - G is the midpoint of side DF - ∠DEG = (3y+ 4)° - ∠FEG = (5y- 10)°
Step 3:: Use the isosceles triangle property
In an isosceles triangle where two sides are equal (DE = EF), the base angles are congruent. This means ∠DEG = ∠FEG.
Step 4:: Set up an equation based on the angle measurements
(3y+4)° = (5y-10)°
Step 5:: Solve for y
y = 7
4 + 10 = 5y - 3y
Step 6:: Calculate ∠DEG
∠DEG = (3(7)+4)° = (21+4)° = 25°
Step 7:: Calculate ∠DEF
Since the base angles of the isosceles triangle are equal, ∠DEF = 2 × ∠DEG = 2(25)° = 50°
Final Answer
m∠DEF = 50°
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