Question

In ΔABC, AB = AD = DC and AB is extended to E. ∠DAC = 22° and ∠E = 24°. Find angles x and y.

Answer 1

To solve for x and y in triangle ABC with the given conditions, we can follow these steps:

Given:

AB = AD = DC, meaning triangle ABD and triangle DCB are isosceles.

AB is extended to E, forming a straight line.

∠DAC = 22° and ∠E = 24° where E is on the extension of AB.

Step 1: Use the Property of Straight Line

Since AB is extended to E, we know that ∠E = 24° an exterior angle at point E.

Since AB is a straight line, we also know that ∠DAB + ∠E = 180° straight line sum.

Thus, 22° + ∠E = 180°

⇒ 22° + 24°

= 180°

This confirms the angles on the straight line add up correctly.

Step 2: Analyze Triangle ABD and Triangle DCB

Since AB = AD = DC, triangles ABD and DCB are both isosceles, meaning:

∠ABD = ∠ADB  ...(Because AB = AD)

∠BDC = ∠DCB  ...(Because DC = DB)

Step 3: Solve for Angles in Triangle ABD

Now, since AB = AD, triangle ABD is isosceles.

Therefore, ∠ABD = ∠ADB = x.

Let’s denote these equal angles by x.

The sum of the angles in triangle ABD is 180°:

∠DAB + ∠ABD + ∠ADB = 180°

Substitute the known values:

22° + x + x = 180°

⇒ 2x = 158°

⇒ x = 79°

Step 4: Solve for Angle y in Triangle DCB

In triangle DCB, since DC = DB, we know triangle DCB is isosceles.

Therefore, ∠DCB = ∠BDC = y.

Let’s denote these equal angles by y.

The sum of the angles in triangle DCB is 180°:

∠DCB + ∠BDC + ∠BCD = 180°.

We already know that ∠BCD = 24° since it is the same as ∠E, which is given as 24°.

Thus, y + y + 24° = 180°

⇒ 2y = 156°

⇒ y = 78°

Final Values:

x = 92°

y = 20°