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Triangle A B C is shown with an angle bisector and two congruent sides. Side A B is congruent to Side A C. Segment A D is the bisector of angle A. Point D is on side B C. Angle B A D is labeled y degrees. Angle B D A is labeled x degrees. Angle A C D is labeled 47 degrees.
Find the values of x and y.
(3 points)
To find the values of x and y, let's use the Angle Bisector Theorem and the Angle Sum Property of a triangle.
Since segment AD is the angle bisector of angle A, we can use the Angle Bisector Theorem to find the ratios between the lengths of BD and DC.
The Angle Bisector Theorem states that the ratio of the lengths of the segments on either side of the angle bisector is equal to the ratio of the lengths of the corresponding opposite sides of the triangle.
So, in triangle ABC, we have:
BD / DC = AB / AC
Since AB = AC (given that side AB is congruent to side AC), the ratio simplifies to:
BD / DC = 1
This means that BD = DC.
Now, let's use the Angle Sum Property of a triangle to find the value of angle BCD.
The Angle Sum Property of a triangle states that the sum of the angles in a triangle is always 180 degrees.
So, in triangle BCD, we have:
angle BCD + angle BDC + angle B = 180 degrees
Since angle BCD is 47 degrees, we can substitute this value into the equation:
47 + angle BDC + angle B = 180
Simplifying the equation, we have:
angle BDC + angle B = 133
Since BD = DC, angles BDC and B are congruent. Let's label both of these angles as y:
y + y = 133
2y = 133
Dividing both sides of the equation by 2, we find:
y = 133 / 2
y = 66.5 degrees
So, the value of angle BAD (x) is equal to 66.5 degrees.
Therefore, the values of x and y are:
x = 66.5 degrees
y = 66.5 degrees
To find the values of x and y, we can use the angle bisector theorem and the fact that the sum of the angles in a triangle is 180 degrees.
1. Angle bisector theorem: According to the angle bisector theorem, the ratio of the lengths of the sides of the triangle formed by the angle bisector is equal to the ratio of the lengths of the sides of the original triangle opposite those angles.
Since AD is the angle bisector of angle A, we can set up the following proportions:
BD/DC = AB/AC
2. Triangle ABC: The triangle ABC has two congruent sides, AB and AC. Therefore, the lengths of AB and AC are equal.
AB = AC
3. Angle sum in a triangle: The sum of the angles in a triangle is always 180 degrees. Therefore, we can write the equation:
x + y + 47 = 180
Now, let's solve for x and y using these equations:
From the angle bisector theorem (1):
BD/DC = AB/AC
Since AB = AC, we can substitute AB for AC in the equation:
BD/DC = AB/AB
BD/DC = 1
Simplifying, we have:
BD = DC
This means that triangle BDA is an isosceles triangle, and the base angles BDA and BAD are congruent. Therefore:
x = y
Now, let's substitute this value of x into the equation from the angle sum in a triangle (3):
x + y + 47 = 180
Substituting x with y gives us:
y + y + 47 = 180
2y + 47 = 180
Simplifying further:
2y = 180 - 47
2y = 133
Divide both sides by 2:
y = 133/2
y = 66.5 degrees
Since x = y, we can conclude that:
x = y = 66.5 degrees.
Therefore, the values of x and y are both 66.5 degrees.
To find the values of x and y, we can use the Angle Bisector Theorem and the fact that the sum of angles in a triangle is equal to 180 degrees.
Step 1: Identify the given angles and sides:
- Angle B A D = y degrees
- Angle B D A = x degrees
- Angle A C D = 47 degrees
- Side A B is congruent to Side A C
Step 2: Apply the Angle Bisector Theorem:
According to the Angle Bisector Theorem, the angle bisector divides the opposite side of the triangle into segments that are proportional to the other two sides.
Using this theorem, we can set up an equation as follows:
A B / B C = A D / D C
Since A B = A C (given), we can substitute A C for A B:
A C / B C = A D / D C
Step 3: Solve for A D:
Cross-multiply the equation:
A C * D C = A D * B C
Since A C = A B and A B = A C (given), we can simplify the equation to:
A B * D C = A D * B C
This equation represents the proportions of the sides and can help us find the value of A D.
Step 4: Find the value of A D:
Substitute the given values into the equation:
A B * D C = A D * B C
Since A B = A C (given), we can substitute A C for A B:
A C * D C = A D * B C
Since A D is the bisector of angle A, we know that angle B A D and angle C A D are congruent. Therefore, angle B C D is also 47 degrees. So, we can substitute 47 degrees for angle C A D:
A C * D C = A D * B C
A C * D C = A D * A C
Divide both sides by A C:
D C = A D
So, we have found that D C is congruent to A D.
Step 5: Find the values of x and y:
We know that angle A C D is 47 degrees.
Since angle A D C is congruent to angle B D A (opposite angles are congruent when a line is drawn through the vertex of an angle bisector), we can conclude that angle B D A is also 47 degrees.
Therefore, x = 47 degrees.
To find the value of y, we can use the fact that the sum of angles in a triangle is 180 degrees.
We know that angle A B D is y degrees, angle B A D is y degrees, and angle B D A is x degrees.
So, y + y + x = 180 degrees.
Since x = 47 degrees, we substitute it into the equation:
y + y + 47 = 180
2y + 47 = 180
Subtract 47 from both sides:
2y = 180 - 47
2y = 133
Divide both sides by 2:
y = 133 / 2
y = 66.5
Thus, the values of x and y are x = 47 degrees and y = 66.5 degrees.