how to find the missing angle of a triangle

Angles In A Triangle

A triangle is the simplest possible polygon. It is a two-dimensional (flat) shape with three straight sides forming an interior, airtight infinite. It has 3 interior angles. I of the earliest concepts to learn in geometry is that triangles have interior angles adding up to $180°$. Just how do you know? How can you prove this is truthful? Permit's detect out!

1. Angles In A Triangle
2. How To Find The Angle of a Triangle
• How To Find The Missing Angle
3. Triangle Angle Formula
4. Angles In A Triangle Sum To 180° Proof

How To Find The Bending of a Triangle

You may have a triangle where but 2 angles take been labelled and measured. Now that y'all are certain all triangles take interior angles adding to $180°$, you tin chop-chop calculate the missing measurement. You can do this one of 2 means:

1. Subtract the ii known angles from $180°$.
2. Plug the two angles into the formula and employ algebra: $a+b+c=180°$

How To Find The Missing Angle of a Triangle

Two known angles of a triangle are $37°$ and $24°$. What is the missing angle?

We can use two different methods to find our missing angle:

1. Subtract the ii known angles from $180°$:

$180°-37°=143°$

$143°-24°=119°$

$c=119°$

2. Plug the two angles into the formula and employ algebra: $a+b+c=180°$

$37°+24°+c=180°$

$61°+c=180°$

$c=119°$

Triangle Angle Formula

Let's draw a triangle and label its interior angles with 3 letters $a$, $b$, and $c$. Our sample will have side $ac$ horizontal at the bottom and $\angle b$ at the tiptop.

Now that we've labeled our angles, we have a formula we tin refer to for the angles. It is $a+b+c=180°$, which tells usa that if we add up all of our angles, they will always equal 180.

Now, let'due south depict a line parallel to side $ac$ that passes through $Pointb$ (which is likewise where you find $\angle b$).

That new parallel line created 2 new angles on either side of $\angle b$. We will label these two angles $\angle z$ and $\angle w$ from left to right. Side $ab$ of our triangle can at present be viewed as a transversal, a line cutting beyond the two parallel lines.

Alternate Interior Angles Theorem

By the Alternate Interior Angles Theorem, we know that $\angle a$ is congruent (equal) to $\angle z$, and $\angle c$ is congruent to $\angle w$.

Did we lose yous? Do non despair! The Alternate Interior Angles Theorem tells usa that a transversal cutting across two parallel lines creates congruent alternating interior angles. Alternate interior angles lie betwixt the parallel lines, on opposite sides of the transversal. In our example, $\angle a$ and $\angle z$ are alternate interior angles, and so are $\angle c$ and $\angle w$.

We now have the 3 angles of our triangle advisedly redrawn and sharing $Pointb$ every bit a common vertex. We have $\angle z$ equally a stand-in for $\angle a$, and so $\angle b$, and finally $\angle w$ as a stand-in for $\angle c$. And look, they form a direct line!

A straight line measures $180°$. This is the same blazon of proof as the parallel lines proof. The three angles of any triangle always add up to $180°$, or a direct line.

Triangle Angle Sum Theorem

Our formula for this is $a+b+c=180°$ where $a$, $b$, and $c$ are the interior angles of any triangle.

Angles In A Triangle Sum To 180° Proof

You lot need four things to do this amazing mathematics trick. You lot need a straightedge, pair of scissors, paper, and pencil. Draw a great, large triangle on a piece of paper. Any triangle -- scalene, isosceles, equilateral, astute, obtuse -- whatever you like.

Label the inside corners (the vertices that form interior angles) with 3 letters, like $R-A-T$. Cut the triangle out, leaving a little border around it so you tin still see all three edges

Now tear off the three corners of your triangle. Do non use the scissors, because you desire jagged edges, which aid you lot avoid confusing them with the direct sides yous drew. Y'all will have three smaller triangular bits, each with an interior angle labelled $R,A$ or $T$. Each little piece has 2 bang-up sides and a crude edge.

You volition besides take a rough hexagon that is the leftover part of the original, larger triangle.

Take your three lilliputian labelled corners and arrange them together so the rough-cutting edges are all away from you lot. The simply way to do that is to make them line upward, to form a straight line. The three interior angles, $RAT$, accept added upward to make a straight bending, also called a straight line.

There; you did it!

Lesson Summary

If you carefully studied this lesson, you lot at present are able to identify and label the iii interior angles of any triangle, and you can retrieve that the interior angles of all triangles add to $180°$. You tin also demonstrate a proof of the sum of interior angles of triangles and utilise a formula, $a+b+c=180°$, where $a$, $b$, and $c$ are the interior angles of the triangle. Further, you tin can calculate the missing measurement of whatsoever interior bending of any triangle using two different methods.

Adjacent Lesson:

Sum of Interior & Outside Angles

Source: https://tutors.com/math-tutors/geometry-help/how-to-find-the-angle-of-a-triangle

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