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One angle of a triangle measures $110^\circ$ . The other two angles are in a ratio of $2:5$ . What are the measures of those two angles? $\newline$ $\underline{\hspace{1cm}}^\circ$ and $\underline{\hspace{1cm}}^\circ$

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Find missing angles in triangles

Full solution

Q. One angle of a triangle measures

110^\circ

. The other two angles are in a ratio of

2:5

. What are the measures of those two angles?

\newline

\underline{\hspace{1cm}}^\circ

and

\underline{\hspace{1cm}}^\circ

Identify Known and Unknown: First, let's identify what we know and what we need to find out. We know that one angle of the triangle measures $110^\circ$ , and we need to find the measures of the other two angles, which are in a ratio of $2:5$ . Since the sum of angles in any triangle is $180^\circ$ , we can set up an equation to find the values of the other two angles.
Set Up Equation: Let's denote the smaller angle as $2x$ and the larger angle as $5x$ . According to the angle sum property of a triangle, the sum of the angles should be $180^\circ$ . Therefore, we can write the equation as: $\newline$ $110^\circ + 2x + 5x = 180^\circ$ .
Combine Like Terms: Now, let's combine like terms ( $2x$ and $5x$ ) to simplify the equation: $\newline$ $110^\circ + 7x = 180^\circ.$ $\newline$ Next, we will subtract $110^\circ$ from both sides to solve for $7x$ : $\newline$ $7x = 180^\circ - 110^\circ.$
Subtract to Solve for x: Performing the subtraction, we get: $\newline$ $7x = 70^\circ$ . $\newline$ Now, we will divide both sides by $7$ to find the value of $x$ : $\newline$ $x = \frac{70^\circ}{7}$ .
Divide to Find x: Calculating the value of x, we get: $\newline$ $x = 10^\circ$ . $\newline$ Now that we have the value of x, we can find the measures of the other two angles by multiplying x by the respective ratio numbers: $\newline$ $2x = 2 \times 10^\circ = 20^\circ$ , $\newline$ $5x = 5 \times 10^\circ = 50^\circ$ .

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