In the triangle, if AB = AC and ∠ABC = 72°, then ∠BAC is:
36°
Understanding the Isosceles Triangle Problem
The question asks us to find the measure of angle ∠BAC in a triangle ABC. We are given that two sides, AB and AC, are equal in length, and one angle, ∠ABC, measures 72°.
A triangle with two sides of equal length is called an isosceles triangle. A key property of isosceles triangles is that the angles opposite the equal sides are also equal.
Applying Isosceles Triangle Properties
In triangle ABC, we are given that AB = AC.
- Side AB is opposite to angle ∠ACB.
- Side AC is opposite to angle ∠ABC.
Since AB = AC, the angles opposite these sides must be equal. Therefore, ∠ACB = ∠ABC.
We are given that ∠ABC = 72°. So, ∠ACB must also be 72°.
Let's summarize the known angles:
| Angle | Measure |
|---|---|
| ∠ABC | $72°$ |
| ∠ACB | $72°$ |
Calculating Angle BAC using Angle Sum Property
The sum of the interior angles in any triangle is always 180°. For triangle ABC, this means:
$\text{∠BAC} + \text{∠ABC} + \text{∠ACB} = 180°$
Now, substitute the known values for ∠ABC and ∠ACB:
$\text{∠BAC} + 72° + 72° = 180°$
Combine the angles:
$\text{∠BAC} + 144° = 180°$
To find ∠BAC, subtract 144° from 180°:
$\text{∠BAC} = 180° - 144°$
$\text{∠BAC} = 36°$
Conclusion: Finding the Angle BAC
Based on the properties of isosceles triangles and the angle sum property of triangles, the measure of angle ∠BAC is 36°.
Revision Table: Key Concepts for Triangle Angles
| Concept | Description | Application |
|---|---|---|
| Isosceles Triangle Property | If two sides are equal, angles opposite them are equal. | AB = AC $\implies$ ∠ABC = ∠ACB |
| Triangle Angle Sum Property | Sum of all angles in a triangle is $180°$. | ∠BAC + ∠ABC + ∠ACB = $180°$ |
Additional Information: Types of Triangles and Angles
Triangles can be classified by their sides and angles.
Classification by Sides:
- Equilateral Triangle: All three sides are equal. All three angles are equal ($60°$ each).
- Isosceles Triangle: Two sides are equal. The angles opposite the equal sides are equal.
- Scalene Triangle: All three sides are different lengths. All three angles are different measures.
Classification by Angles:
- Acute Triangle: All three angles are acute (less than $90°$).
- Right Triangle: One angle is a right angle ($90°$).
- Obtuse Triangle: One angle is obtuse (greater than $90°$).
Understanding these classifications helps in solving geometry problems involving triangles.
Important Questions from Lines and Angles
If angles of a triangle are in the ration of 2 : 3 : 4, then the measure of the smallest angle is:
The angles of a triangle are (8x - 15)°,(6x - 11)° and ( 4x – 10)°. What is the value of x ?
In a ΔABC, the bisectors of ∠B and ∠C meet at point O, inside the triangle. If ∠BOC = 122°, then the measure of ∠A is:
In ΔABC, D is a point on side BC such that ∠ADC = 2∠BAD. If ∠A = 80° and ∠C = 38°, then what is the measure of ∠ADB?
In ∆ABC, ∠B = 68° and ∠C = 32°. Sides AB and AC are produced to points D and E respectively. The bisectors of ∠DBC and ∠BCE meet at F. what is the measure of ∠BFC?
