Solving Triangle Angles: Finding ∠ADB

We are given a triangle ΔABC, with a point D on side BC. We are provided with the following information:

• ∠A = 80°
• ∠C = 38°
• ∠ADC = 2∠BAD

Our goal is to find the measure of ∠ADB.

Step 1: Find ∠B in ΔABC

The sum of angles in any triangle is 180°. In ΔABC, we have:

$\angle A + \angle B + \angle C = 180^\circ$

Substituting the given values for ∠A and ∠C:

$80^\circ + \angle B + 38^\circ = 180^\circ$

$118^\circ + \angle B = 180^\circ$

$\angle B = 180^\circ - 118^\circ$

$\angle B = 62^\circ$

Step 2: Define Variables based on the Given Relation

We are given that ∠ADC = 2∠BAD. Let's assign a variable to ∠BAD.

Let $\angle BAD = x$.

Then, $\angle ADC = 2x$.

Step 3: Relate ∠ADB and ∠ADC

Angles ∠ADB and ∠ADC form a linear pair on the straight line BC. Therefore, they are supplementary.

$\angle ADB + \angle ADC = 180^\circ$

Substituting the expression for ∠ADC:

$\angle ADB + 2x = 180^\circ$

So, $\angle ADB = 180^\circ - 2x$.

Step 4: Use Angle Sum Property in ΔABD

Now consider ΔABD. The sum of its angles is 180°.

$\angle BAD + \angle B + \angle ADB = 180^\circ$

Substitute the expressions and value we found:

$x + 62^\circ + (180^\circ - 2x) = 180^\circ$

Step 5: Solve the Equation for x

Let's simplify and solve the equation from Step 4:

$x + 62^\circ + 180^\circ - 2x = 180^\circ$

Combine like terms:

$(x - 2x) + (62^\circ + 180^\circ) = 180^\circ$

$-x + 242^\circ = 180^\circ$

Subtract 242° from both sides:

$-x = 180^\circ - 242^\circ$

$-x = -62^\circ$

Multiply by -1:

$x = 62^\circ$

Step 6: Calculate ∠ADB

We found that $\angle ADB = 180^\circ - 2x$. Now substitute the value of $x$ we just found:

$\angle ADB = 180^\circ - 2(62^\circ)$

$\angle ADB = 180^\circ - 124^\circ$

$\angle ADB = 56^\circ$

Verification (Optional)

If $\angle BAD = x = 62^\circ$, then $\angle DAC = \angle A - \angle BAD = 80^\circ - 62^\circ = 18^\circ$. $\angle ADC = 2x = 2(62^\circ) = 124^\circ$. $\angle ADB = 56^\circ$.

Check ΔABD: $\angle BAD + \angle B + \angle ADB = 62^\circ + 62^\circ + 56^\circ = 180^\circ$. (Correct)

Check ΔADC: $\angle DAC + \angle C + \angle ADC = 18^\circ + 38^\circ + 124^\circ = 180^\circ$. (Correct)

The values are consistent.

Final Answer

The measure of ∠ADB is 56°.

Revision Table: Key Geometry Concepts

Additional Information: Geometry Problem Solving

Solving geometry problems often involves using fundamental theorems and properties of shapes. For triangle problems, common techniques include:

• Using the angle sum property.
• Identifying linear pairs and supplementary angles.
• Using exterior angle theorem (the exterior angle of a triangle is equal to the sum of the two opposite interior angles). In this problem, $\angle ADC$ is an exterior angle to ΔABD at vertex D. So, $\angle ADC = \angle BAD + \angle B$. This would give $2x = x + 62^\circ$, leading to $x = 62^\circ$, which is an alternative way to solve for $x$.
• Using properties of specific triangles (like isosceles, equilateral, right triangles) if applicable.
• Setting up equations based on given angle relationships.
• Breaking down complex figures into simpler triangles.

Always draw a diagram and label the given information. This helps visualize the relationships between angles and sides. Double-checking your calculations and verifying the solution by plugging the found values back into the original conditions is a good practice.