Angles in Triangles

Given any triangle, the sum of the measures of the interior angles is equal to 180^(∘). This is what the Triangle Interior Angles Theorem says. This means that the missing part of the given statement is 180^(∘).

The sum of the measures of the interior angles of a triangle is 180^(∘).

We can justify this statement algebraically. Let's consider a triangle ABC.

Let's draw a line parallel to BC passing through A.

Notice that AC and AB are transversals that cut a pair of parallel lines. As a result, the alternate interior angles are congruent. This means that ∠ B ≅ ∠ DAB and ∠ C ≅ ∠ EAC.

The diagram shows that ∠ DAB, ∠ BAC, and ∠ EAC form a straight angle. This means that the sum of their measures is equal to 180^(∘). m∠ EAB + m∠ BAC + m∠ DAC = 180^(∘) We can substitute m∠ B for m∠ EAB and m∠ C for m∠ DAC. This is because congruent angles have the same measure. m∠ EAB+ m∠ BAC + m∠ DAC = 180^(∘) ⇓ m∠ B + m∠ BAC + m∠ C= 180^(∘) ✓ The last equation tells us that the sum of the measures of the interior angles of a triangle is equal to 180^(∘).

Let's draw a triangle and one of its exterior angles.

Next, let's consider the interior angle that is adjacent to ∠ SQR. In other words, let's consider ∠ PQR.

The diagram shows that the exterior angle and its corresponding interior angle form a linear pair, which means that their measures add up to 180^(∘). As a result, they form a straight angle. This is the phrase that fills in the blank!

Each exterior angle forms a straight angle with its corresponding interior angle.

Let's begin by recalling what an exterior angle of a triangle is.

An exterior angle is the angle formed between one side of a triangle and the extension of an adjacent side.

We can see that the sides of ∠ 2 are extensions of two sides of the triangle. In other words, none of the sides of ∠ 2 is a side of the triangle. The same situation happens with ∠ 3. This means that neither ∠ 2 nor ∠ 3 are exterior angles of the triangle.

On the other hand, we can see that ∠ 1, ∠ 4, and ∠ 5 all are formed by one side of the triangle and the extension of an adjacent side. As a result, we can conclude that ∠ 1, ∠ 4, and ∠ 5 are all exterior angles of the given triangle.

Notice that ∠ 3 and ∠ 5 are both formed by the extensions of the sides of the triangle. In other words, none of these angles is formed by one side of the triangle. This means that none of these angles is an exterior angle of the triangle.

We need to be careful about ∠ 1. As shown, it is formed by one side of the triangle and the extension of the same side. This is not what the definition says. It should be formed by one side of the triangle and the extension of an adjacent side to be an exterior angle. We then conclude that ∠ 1 is not an exterior angle.

On the contrary, ∠ 2, ∠ 4, and ∠ 6 are all formed by one side of the triangle and the extension of an adjacent side. This means that these three angles are exterior angles of the triangle.

Let's begin by identifying the exterior angle of the triangle. An exterior angle is formed by one side of the triangle and an extension of an adjacent side. Also, this angle is outside the triangle. In the given diagram, ∠ 4 meets these conditions.

Now that we identified the exterior angle, let's recall the definition of a remote interior angle.

A remote interior angle is an interior angle that is not adjacent to a specific exterior angle.

Of the three interior angles of the triangle, ∠ 1 is adjacent to ∠ 4. This means that ∠ 1 is not a remote interior angle corresponding to ∠ 4. On the contrary, ∠ 2 and ∠ 3 meet the conditions in the definition and so, these two angles are remote interior angles corresponding to ∠ 4.

We can identify ∠ 1 as an exterior angle because it is outside the triangle and is formed by one side of the triangle and an extension of an adjacent side.

Recall, a remote interior angles is an interior angle that is not adjacent to a specific exterior angle. In our case, the remote interior angles corresponding to ∠ 1 are ∠ 2 and ∠ 4.

We are given a triangle and the measures of two interior angles. Our job is finding the third angle measure. Recall, the sum of the measures of the interior angles of a triangle is 180^(∘). Let's write an equation using this fact. m∠ A + m∠ B + m∠ C = 180^(∘) From the diagram, we have that m∠ B=33^(∘) and m∠ C = 37^(∘). Let's substitute these measures into the equation and solve it for m∠ A.

The missing angle measure is 110^(∘).

In △ JKL, we were given the measure of ∠ K and have to find the measure of ∠ L. This seems like insufficient information but, notice that ∠ J is a right angle which means that it measures 90^(∘).

Let's write an equation relating the three angle measures using the Interior Angles Theorem. From the obtained equation, we can find the missing angle measure.

In △ MNO, the angles N and O are remote interior angles corresponding to the exterior angle PMN. Recall that there is a close relationship between these three angles.

Triangle Exterior Angle Theorem |- The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote angles.

Let's write an equation using this information. m∠ PMN = m∠ N + m∠ O Next, we substitute 25^(∘) for m∠ N and 35^(∘) for m∠ O. m∠ PMN &= 25^(∘) + 35^(∘) &⇓ m∠ PMN &= 60^(∘) Finally, let's update the given diagram and write the measure of the exterior angle.

Notice that ∠ EFG is an exterior angle, and ∠ D and ∠ E are its two corresponding remote interior angles.

We can write an equation relating the measures of these three angles using the Triangle Exterior Angle Theorem. m∠ EFG = m∠ D + m∠ E From the diagram, ∠ D is a right angle and so its measure is 90^(∘). Also, m∠ E = 53^(∘). Let's substitute these values into the equation to find the measure of the exterior angle. m∠ EFG &= 90^(∘) + 53^(∘) &⇓ m∠ EFG &= 143^(∘)