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 $75^\circ$.
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Important Questions from Geometry
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Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
