Angles in Triangles

The sum of the angles in a triangle

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 $\cl"red"{∠B}$.

Does changing the length of $\cl"red"\ov{AB}$ (sliding the $\cl"red"c$ slider) affect the measure of $\cl"red"{∠C}$?

Does changing the measure of $\cl"red"{∠A}$ affect the measure of $\cl"red"{∠C}$?

Does changing the measure of $\cl"red"{∠B}$ affect the measure of $\cl"red"{∠C}$?

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.

measure of $\cl"red"{∠A}$	measure of $\cl"red"{∠B}$	measure of $\cl"red"{∠C}$	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 $\cl"blue"{DE}$.

As you slide the sliders, notice that the marked angle $\cl"blue"{∠ECF}$ always has the same measure as one of the triangle’s angles ($\cl"red"{∠BAC}$, $\cl"red"{∠CBA}$, and $\cl"red"{∠ACB}$). Which of these three angles is it? (Now that we’ve started drawing other angles which have the points $\cl"red"A$, $\cl"red"B$, and $\cl"red"C$ as vertices, we’ll use the full three-letter names $\cl"red"{∠BAC}$, $\cl"red"{∠CBA}$, and $\cl"red"{∠ACB}$ for the angles of $\cl"red"{▵ABC}$ to avoid confusion.)

Does $\cl"blue"{∠ECF}$ have the same measure as that angle because they are vertical angles, because they are corresponding angles, or for neither reason?

Click to see $\cl"red"{▵ABC}$ along with the same two parallel lines, but with a different triangle side extended into the transversal $\cl"blue"{BC}$.

As you slide the sliders, notice that the new marked angle $\cl"blue"{∠GCD}$ always has the same measure as one of the triangle’s angles ($\cl"red"{∠BAC}$, $\cl"red"{∠CBA}$, and $\cl"red"{∠ACB}$). Which of these three angles is it?

Does $\cl"blue"{∠GCD}$ have the same measure as that angle because they are vertical angles, because they are corresponding angles, or for neither reason?

Click to see $\cl"red"{▵ABC}$ with all of its sides extended, but with no parallel line drawn.

The angle $\cl"blue"{∠FCG}$ is now marked. Notice that as you slide the sliders it always has the same measure as one of the triangle’s angles ($\cl"red"{∠BAC}$, $\cl"red"{∠CBA}$, and $\cl"red"{∠ACB}$). Which of these three angles is it?

Does $\cl"blue"{∠FCG}$ have the same measure as that angle because they are vertical angles, because they are corresponding angles, or for neither reason?

Now all of the angles from one-angle-qn are drawn at once.

The three angles $\cl"blue"{∠ECF}$, $\cl"blue"{∠FCG}$, and $\cl"blue"{∠GCD}$ can be put together to form a straight line. What is the sum of their measures?

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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 $\cl"red"{∠CBA}$.

What is the sum of the measures of the angles of $\cl"red"{▵ABC}$?

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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.

What is the measure of $\cl"red"{∠A}$?

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Exterior angles and their sum

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.

measure of $\cl"purple"{∠DAB}$	measure of $\cl"green"{∠EBC}$	measure of $\cl"orange"{∠FCA}$	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 $\cl"blue"{CG}$.

Notice that the newly drawn green angle $∠GCF$ has the same measure as the green exterior angle $∠EBC$. Is this because they are vertical angles, because they are corresponding angles, or for neither reason?

Click to see the side $\cl"red"\ov{AC}$ extended into a transversal of the parallel lines $\cl"blue"{AB}$ and $\cl"blue"{CG}$.

Again, notice that the newly drawn purple angle $∠ACG$ has the same measure as the purple exterior angle $∠DAB$. Is this because they are vertical angles, because they are corresponding angles, or for neither reason?

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 $\cl"red"C$.

There are 360 degrees in a complete circle. Since the three angles $\cl"green"{∠GCF}$, $\cl"purple"{∠ACG}$, and $\cl"orange"{∠FCA}$ form a complete circle around $\cl"red"C$, what does the sum of their measures have to be?

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In partial-exterior-qn, you saw that $\cl"green"{∠GCF}$ always has the same measure as the exterior angle $\cl"green"{∠EBC}$, and $\cl"purple"{∠ACG}$ always has the same measure as the exterior angle $\cl"purple"{∠DAB}$. What is the sum of the measures of the three exterior angles $\cl"purple"{∠DAB}$, $\cl"green"{∠EBC}$, and $\cl"orange"{∠FCA}$?

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As you can see:

The measures of three exterior angles at distinct vertices of a triangle add up to 360˚.