Question Video: Calculating the Internal Angle Measures in a Given Triangle | Nagwa Question Video: Calculating the Internal Angle Measures in a Given Triangle | Nagwa

Question Video: Calculating the Internal Angle Measures in a Given Triangle Mathematics • First Year of Preparatory School

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In the figure, find the measure of all the angles in triangle 𝐴𝐡𝐢.

03:50

Video Transcript

In the following figure, find the measure of all the angles in triangle 𝐴𝐡𝐢.

In this question, we’re given a figure containing expressions for the measure of the internal angles in a triangle 𝐴𝐡𝐢 in terms of an unknown π‘₯. We need to determine the measures of all three of these angles. To do this, we can start by recalling that the sum of all of the internal angles’ measures in a triangle is 180 degrees. Therefore, we have that the measure of angle 𝐴 plus the measure of angle 𝐡 plus the measure of angle 𝐢 is 180 degrees. We can substitute the given expressions for the angle measures into this equation and remove the units to obtain that π‘₯ plus one plus three π‘₯ minus one plus three π‘₯ is equal to 180.

To solve this equation for π‘₯, we want to isolate π‘₯ on one side of the equation. To do this, let’s combine the like terms to get seven π‘₯ and then calculate that one minus one is zero. So, we have seven π‘₯ equals 180. We can then solve for π‘₯ by dividing both sides of the equation by seven. We get that π‘₯ equals 180 over seven. We cannot simplify this any further since seven and 180 share no common factors above one.

We can use this value of π‘₯ to find the measures of the three internal angles by substituting them into the given expressions for the measure and evaluating. First, we substitute π‘₯ equals 180 over seven into the expression for the measure of angle 𝐴. We get the measure of angle 𝐴 equals 180 over seven plus one. We can evaluate this expression by rewriting one as seven over seven. We get that the measure of angle 𝐴 is 187 over seven degrees.

We can follow the same process for the measure of angle 𝐡. We have that the measure of angle 𝐡 equals three times 180 over seven minus one. We can then evaluate this expression by multiplying the numerator by three and rewriting one as seven over seven. We get that the measure of angle 𝐡 is 533 over seven degrees. Finally, substituting π‘₯ equals 180 over seven into the expression for the measure of angle 𝐢 and evaluating gives us that the measure of angle 𝐢 is 540 over seven degrees.

It can be useful to check that the sum of these angle measures is 180 degrees. If we added these measures together, we would obtain 187 plus 533 plus 540 all over seven, which is 180 degrees. Hence, the measure of angle 𝐴 is 187 over seven degrees. The measure of angle 𝐡 is 533 over seven degrees. And the measure of angle 𝐢 is 540 over seven degrees.

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