in an isosceles trapezoid, ABCD, AD congruent BC. diagonal AC bisects angle DAB. if the measure of angle CAB = 15, find: the measure of angle b, measure of angle BCA, measure of ACD, and the measure of angle D
∠A = 15+15 = 30 = ∠B
∠C = ∠D = (360-30-30)/2 = 150
so, ∠ACD = 180-150-15 = 15
∠BCA = 150-15 = 135
Given m = 10 and n = 9, the ratio of n to m is 10/9.
True
False
Let's break down the problem step-by-step:
Step 1: Draw a diagram.
Draw an isosceles trapezoid ABCD, with AD congruent to BC. Label the diagonals AC and BD. Angle CAB should be 15 degrees, and angle DAB should also be 15 degrees.
Step 2: Use the fact that an isosceles trapezoid has congruent base angles.
Since AD is congruent to BC and angles DAB and BCA are base angles, we know that angle DAB is congruent to angle BCA.
Step 3: Determine the measure of angle BCA.
Since angle CAB is 15 degrees and angle DAB is congruent to BCA, angle BCA is also 15 degrees.
Step 4: Use the fact that the diagonal AC bisects angle DAB.
Since AC bisects angle DAB, we know that angle CAD is congruent to angle BAD.
Step 5: Determine the measure of angle CAD.
Since angle CAB is 15 degrees, and angle CAD is congruent to angle BAD, angle CAD is also 15 degrees.
Step 6: Use the fact that angles in a triangle add up to 180 degrees.
Since triangle CAD has angles of 15 degrees, 15 degrees, and 180 - (15 + 15) = 150 degrees, we can conclude that angle ACD measures 150 degrees.
Step 7: Determine the measure of angle D.
Since the sum of all angles in a quadrilateral equals 360 degrees, and we know that angle ACD is 150 degrees, we subtract that from 360 to find angle D. Therefore, angle D measures 360 - 150 = 210 degrees.
To summarize:
- The measure of angle B is 15 degrees.
- The measure of angle BCA is 15 degrees.
- The measure of angle ACD is 150 degrees.
- The measure of angle D is 210 degrees.
To find the measures of the angles in the isosceles trapezoid ABCD, we can use the properties of isosceles trapezoids and the given information.
1. Measure of angle B:
Since diagonal AC bisects angle DAB, we know that angle CAB is congruent to angle BAC. Therefore, angle CAB = angle BAC = 15 degrees.
2. Measure of angle BCA:
In an isosceles trapezoid, the base angles (angles adjacent to the non-parallel sides) are congruent. Therefore, angle BCA = angle CAB = 15 degrees.
3. Measure of angle ACD:
In an isosceles trapezoid, the base angles are congruent. So angle ACD is congruent to angle BDC. Since AD is congruent to BC, the angles opposite those sides will also be congruent. Therefore, angle ACD = angle BDC.
4. Measure of angle D:
The sum of the measures of the angles in a trapezoid is 360 degrees. In this case, we can find the measure of angle D by subtracting the measures of angle BCA, angle CAB, and angle ACD from 360 degrees.
Angle D = 360 degrees - angle BCA - angle CAB - angle ACD
Now, let's substitute the given values:
Angle D = 360 degrees - 15 degrees - 15 degrees - angle ACD
Therefore, we need to know the measure of angle ACD in order to find the measure of angle D.