Understanding the Triangle Angle Problem

This problem involves finding the measure of a specific angle (∠ADC) within a triangle (ΔABC), given some information about other angles and a point D on one side (BC). We are given the measures of two angles of ΔABC, and a relationship between the angle ∠ADB and ∠DAC.

Given Information:

• In ΔABC, D is a point on BC.
• The relationship: $\angle \text{ADB} = 2 \angle \text{DAC}$.
• ∠BAC = 70°.
• ∠B = 56°.

Goal:

Find the measure of ∠ADC.

Step-by-Step Solution

Step 1: Find ∠C in ΔABC

The sum of angles in any triangle is 180°. In ΔABC, we know ∠BAC and ∠B. We can find ∠C using the angle sum property:

$\angle \text{BAC} + \angle \text{B} + \angle \text{C} = 180^\circ$

Substitute the given values:

$70^\circ + 56^\circ + \angle \text{C} = 180^\circ$

$126^\circ + \angle \text{C} = 180^\circ$

$\angle \text{C} = 180^\circ - 126^\circ$

$\angle \text{C} = 54^\circ$

So, the measure of angle C is 54°.

Step 2: Use the Exterior Angle Property in ΔADC

Consider the triangle ΔADC. The angle ∠ADB is an exterior angle to ΔADC because D is on the line segment BC, and A and C are vertices of the triangle.

The exterior angle property states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles.

For ΔADC, the exterior angle ∠ADB is equal to the sum of the opposite interior angles ∠DAC and ∠C.

$\angle \text{ADB} = \angle \text{DAC} + \angle \text{C}$

Step 3: Set up an Equation and Solve for ∠DAC

We are given that $\angle \text{ADB} = 2 \angle \text{DAC}$. Let's denote $\angle \text{DAC}$ as $x$. Then $\angle \text{ADB} = 2x$. We also found that $\angle \text{C} = 54^\circ$.

Substitute these into the exterior angle equation from Step 2:

$2x = x + 54^\circ$

Subtract $x$ from both sides:

$2x - x = 54^\circ$

$x = 54^\circ$

So, $\angle \text{DAC} = 54^\circ$.

Step 4: Calculate ∠ADB

We know that $\angle \text{ADB} = 2 \angle \text{DAC}$ and we found $\angle \text{DAC} = 54^\circ$.

$\angle \text{ADB} = 2 \times 54^\circ$

$\angle \text{ADB} = 108^\circ$

Step 5: Calculate ∠ADC

Angles ∠ADB and ∠ADC lie on the straight line BC and share the vertex D. They form a linear pair. The sum of angles in a linear pair is 180°.

$\angle \text{ADB} + \angle \text{ADC} = 180^\circ$

Substitute the value of ∠ADB we just found:

$108^\circ + \angle \text{ADC} = 180^\circ$

Subtract 108° from both sides:

$\angle \text{ADC} = 180^\circ - 108^\circ$

$\angle \text{ADC} = 72^\circ$

Thus, the measure of ∠ADC is 72°.

Summary of Angles Calculated

Revision Table: Key Concepts

Additional Information: Triangle Angle Properties

Understanding angle properties is crucial for solving geometry problems involving triangles. Here are a few more points:

• Interior Angles: Angles inside the triangle. A triangle always has three interior angles.
• Exterior Angles: Formed by extending one side of the triangle. There are six exterior angles (two at each vertex), but they come in pairs of vertical angles. The exterior angle at a vertex and the interior angle at the same vertex form a linear pair (sum is 180°).
• Types of Triangles based on Angles:
  • Acute Triangle: All three interior angles are less than 90°.
  • Right Triangle: One interior angle is exactly 90°. The other two angles are complementary (sum to 90°).
  • Obtuse Triangle: One interior angle is greater than 90°. The other two angles are acute.

Mastering these basic properties will help you solve a wide range of geometry questions about angles in triangles.