Given: obtuse triangle ABC with obtuse angle B. Point D on side AC such that angle BDC is also obtuse. Angle DAB is 2/3 of angle ABD. Angle BCD is 1/5 angle BDC and angle CBD is 4 more than angle BAD. What is the measure of angle ABC?

Question

Given: obtuse triangle ABC with obtuse angle B. Point D on side AC such that angle BDC is also obtuse. Angle DAB is 2/3 of angle ABD. Angle BCD is 1/5 angle BDC and angle CBD is 4 more than angle BAD. What is the measure of angle ABC?

Answer 1

½ m ےABC =½ m ےPQR

Answer 2

To find the measure of angle ABC, we need to analyze the given information step by step and use the properties of triangles and angles.

Let's denote the measure of angle BAD as x (in degrees).

According to the given information, angle DAB is 2/3 of angle ABD. This means that angle DAB measures (2/3)x degrees.

Since angle BCD is 1/5 of angle BDC, we can express angle BCD as (1/5)(180 - angle BCD) degrees. Since angles BCD and angle BDC add up to 180 degrees (because it's a straight line), we can simplify this to (1/5)(180 - angle BDC) degrees.

The next statement tells us that angle CBD is 4 more than angle BAD, so angle CBD measures x + 4 degrees.

Now, let's sum up the angles of triangle ABC:

angle A + angle B + angle C = 180 degrees

Since we know that angle DAB and angle ABD are both part of angle A, we can rewrite it as:

(2/3)x + x + angle B + (1/5)(180 - angle BDC) = 180

We can simplify this equation by combining like terms:

(5/3)x + angle B + 36 - (1/5)angle BDC = 180

Now, let's substitute the value of angle BDC into the equation. Recall that angle BDC is obtuse, which means it measures more than 90 degrees.

From earlier, we know that angle DAB is 2/3 of angle ABD:

2x/3 + x + angle B + (1/5)(180 - angle BDC) = 180

Simplifying further:

5x/3 + angle B + 36 - (1/5)(180 - angle BDC) = 180

Now, substitute the value of angle CBD (x + 4 degrees) into the equation:

5x/3 + angle B + 36 - (1/5)(180 - (x + 4)) = 180

Simplifying once more:

5x/3 + angle B + 36 - (1/5)(176 - x) = 180

Now, combine like terms and solve for the value of x:

5x/3 + angle B + 36 - (176 - x)/5 = 180

Multiply through by 15 to eliminate the fractions:

25x + 15(angle B) + 540 - 3(176 - x) = 2700

25x + 15(angle B) + 540 - 528 + 3x = 2700

Combine like terms:

28x + 15(angle B) + 12 = 2700

28x + 15(angle B) = 2688

Now, it seems that we need more information to determine the specific values of x and angle B. The given information does not provide enough details to solve for these values. Therefore, it is not possible to determine the measure of angle ABC without additional information.