Exterior Angle Theorem Application
In triangle ABC, the side BC is extended to point D. The exterior angle at C is given as $105^\circ$. The interior angles at A and B are in the ratio $2:5$. Let the measures of interior angles A and B be $2x$ and $5x$, respectively.
Exterior Angle Calculation
According to the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Exterior angle at C = Angle A + Angle B
$105^\circ = 2x + 5x$
$105^\circ = 7x$
Solving for $x$:
$x = \frac{105^\circ}{7}$
$x = 15^\circ$
Interior Angle Measures
Now, we can find the measures of angles A and B:
- Angle A = $2x = 2 \times 15^\circ = 30^\circ$
- Angle B = $5x = 5 \times 15^\circ = 75^\circ$
The sum of the interior angles of a triangle is $180^\circ$. Therefore, the interior angle at C is:
Angle C = $180^\circ - (\text{Angle A} + \text{Angle B})$
Angle C = $180^\circ - (30^\circ + 75^\circ)$
Angle C = $180^\circ - 105^\circ = 75^\circ$
Largest Interior Angle Identification
The interior angles of triangle ABC are $30^\circ$, $75^\circ$, and $75^\circ$. The largest interior angle among these is $75^\circ$.
Similar Questions
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Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
