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In this lesson, you will learn some relationships among the angles of a triangle. They will build upon the facts you’ve already learned about supplementary angles, vertical angles, and corresponding angles.
A triangle $\cl"red"{▵ABC}$ is drawn to the left, along with sliders that let you change the length of the side $\cl"red"\ov{AB}$ and the measures of the angles $\cl"red"{∠A}$ and $\cl"red"{∠B}$.
Each row of the table below gives measures for $\cl"red"{∠A}$ and $\cl"red"{∠B}$. Slide the sliders to find the corresponding measure of $\cl"red"{∠C}$, and the sum of all three measures.
In the figure to the left, you can see the same triangle $\cl"red"{▵ABC}$ drawn on top of the lines $\cl"blue"{AB}$ and $\cl"blue"{AC}$, as well as the line $\cl"blue"{DE}$ passing through $\cl"red"C$ that is parallel to $\cl"blue"{AB}$. So $\cl"blue"{AC}$ is a transversal intersecting the parallel lines $\cl"blue"{AB}$ and $\cl"blue"{DE}$.
Click to see $\cl"red"{▵ABC}$ along with the same two parallel lines, but with a different triangle side extended into the transversal $\cl"blue"{BC}$.
Click to see $\cl"red"{▵ABC}$ with all of its sides extended, but with no parallel line drawn.
Now all of the angles from one-angle-qn are drawn at once.
As you saw in one-angle-qn, each of those three angles has the same measure as one of the angles of $\cl"red"{▵ABC}$: $\cl"blue"{∠ECF}$ has the same measure as $\cl"red"{∠BAC}$, $\cl"blue"{∠FCG}$ has the same measure as $\cl"red"{∠ACB}$, and $\cl"blue"{∠GCD}$ has the same measure as $\cl"red"{∠CBA}$.
As you can see:
The measures of the angles of any triangle add up to 180˚.
A triangle $\cl"red"{▵ABC}$ is drawn to the left, and the measures of two of its angles ($\cl"red"{∠B}$ and $\cl"red"{∠C}$) are given. If the measure of $\cl"red"{∠A}$ in degrees is $x$, use the fact that the measures of all three angles add up to 180˚ to write an equation for $x$. Then simplify that equation into the form $x+m=n$.
Solve the equation from find-angle-sum-qn for $x$ by isolating $x$. Then check your answer.
Each side of the triangle $\cl"red"{▵ABC}$ drawn to the left has been extended into a ray. The angles that are drawn between these rays and each adjacent side of the triangle are called exterior angles of the triangle. (Sometimes the usual angles inside a triangle are called interior angles, to remind you that they aren’t exterior angles.) The sliders let you control the measures of two of these exterior angles, as well as the length of the triangle side $\cl"red"\ov{AB}$.
Each row of the table below gives measures for the exterior angles $\cl"purple"{∠DAB}$ and $\cl"green"{∠EBC}$. Slide the sliders to find the measure of the third exterior angle $\cl"orange"{∠FCA}$, and the sum of all three measures.
In the figure to the left, the line $\cl"blue"{CG}$ has been drawn parallel to the triangle side $\cl"red"\ov{AB}$, and $\cl"red"\ov{BC}$ has been extended into a transversal of the two parallel lines $\cl"blue"{AB}$ and $\cl"blue"{CG}$.
Click to see the side $\cl"red"\ov{AC}$ extended into a transversal of the parallel lines $\cl"blue"{AB}$ and $\cl"blue"{CG}$.
Now both of the new angles $\cl"green"{∠GCF}$ and $\cl"purple"{∠ACG}$ from the previous question are drawn to the left. Notice that, together with the orange exterior angle $∠FCA$, they form a complete circle around the point $\cl"red"C$.
The measures of three exterior angles at distinct vertices of a triangle add up to 360˚.