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 the side DF. Segment EG intersect the side DF at point G. The measure of angle DEG is (3y+4) degrees and the measure of angle FEG is (5y-10) degrees.
The Correct Answer and Explanation is :
The correct answer is: 130 degrees
To find the measure of angle ( m∠DEF ), we start by analyzing the given information about triangle ( DEF ). We know that sides ( DE ) and ( EF ) are congruent, which indicates that triangle ( DEF ) is isosceles. In an isosceles triangle, the angles opposite the congruent sides are equal. Therefore, we can say:
[
m∠DEG = m∠FEG
]
From the problem, we are given:
[
m∠DEG = 3y + 4
]
[
m∠FEG = 5y – 10
]
Since these angles are equal, we can set up the following equation:
[
3y + 4 = 5y – 10
]
Next, we solve for ( y ):
- Subtract ( 3y ) from both sides:
[
4 = 2y – 10
]
- Add ( 10 ) to both sides:
[
14 = 2y
]
- Divide both sides by ( 2 ):
[
y = 7
]
Now that we have the value of ( y ), we can substitute it back into the expressions for ( m∠DEG ) and ( m∠FEG ):
[
m∠DEG = 3(7) + 4 = 21 + 4 = 25 \text{ degrees}
]
[
m∠FEG = 5(7) – 10 = 35 – 10 = 25 \text{ degrees}
]
Since both angles ( DEG ) and ( FEG ) measure ( 25 ) degrees, we can find ( m∠DEF ) using the fact that the sum of angles in triangle ( DEF ) must equal ( 180 ) degrees.
Let ( m∠DEF = x ). Thus:
[
m∠DEF + m∠DEG + m∠FEG = 180
]
Substituting in the known values:
[
x + 25 + 25 = 180
]
This simplifies to:
[
x + 50 = 180
]
Subtracting ( 50 ) from both sides gives:
[
x = 130
]
Therefore, the measure of angle ( m∠DEF ) is
[
\boxed{130 \text{ degrees}}
]
This solution illustrates how we utilized properties of isosceles triangles and algebraic manipulation to determine the measures of the angles within triangle ( DEF ).