Triangle Abc Has The Angle Measures Shown? The 128 Correct Answer

triangle

what is a triangle

A simple closed curve or polygon made up of three line segments (sides) is called a triangle.

The shapes shown above are triangles. The symbol of a triangle is ∆.

A triangle is a polygon with three sides. In the given figure, ABC is a triangle. AB, BC and CA are its sides. The point where two sides meet is called the vertex. A, B, C are its vertices. There are many types of triangles. Triangles can be classified based on sides and angles. Some triangles given below have been classified based on their sides.

has a triangle

three line segments or sides

three corners three angles

There are six types of triangles, 3 relating to sides and 3 relating to angles.

Three types of triangles related to the sides

(i) A triangle in which all three line segments or sides are unequal is called an unequal triangle.

Scalene Triangle

A triangle that has no equal sides is called a scalene triangle. ABC in the given figure is an uneven triangle whose sides AB, BC and CA are of different lengths.

(ii) A triangle whose sides or two line segments are equal is called an isosceles triangle.

Here AB = AC.

isosceles triangle

A triangle that has two equal sides is called an isosceles triangle. ABC in the given figure is an isosceles triangle whose sides AB and AC are equal.

(iii) A triangle where all three line segments or sides are equal is called an equilateral triangle.

Here AB = BC = CA.

Equilateral triangle

A triangle that has three equal sides is called an equilateral triangle. ABC in the given figure is an equilateral triangle whose sides are all equal. AB=BC=CA.

Types of triangles related to angles

(i) A triangle in which all three angles are acute is called an acute triangle.

∠ABC, ∠ACB, and ∠BAC are all acute angles.

Sharp angled triangle

(ii) A triangle in which one of the three angles is a right angle is called a right triangle.

∠ABC = a right angle.

Right triangle

(iii) A triangle in which one of the three angles is more than one right angle (or is an obtuse angle) is called an obtuse triangle.

∠ABC is an obtuse angle.

obtuse triangle

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Geometry related concepts – Simple shapes and circles

● Simple closed curves

● Polygon

● Different types of polygons

● Angle

● triangle

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Angles of a triangle – explanation & examples

We know that every shape in the universe is based on angles. The square is basically four lines connected in such a way that each line forms a 90 degree angle with the other line. In this way, a square has four 90-degree angles on its four sides.

Similarly, a straight line extends 180 degrees on both sides. If it rotates at any point, it becomes two lines separated by a certain angle. In the same way, a triangle is basically three lines that are connected at certain angle values.

These angular measurements define the type of triangle. Therefore, angles are essential in the study of any geometric shape.

In this article, you will learn the angles of a triangle and how to find the unknown angles of a triangle when you only know some of the angles. To know the important concepts of triangles, you can consult the previous articles.

What are the angles of a triangle?

The angle of a triangle is the space formed between two side lengths of a triangle. A triangle contains interior angles and exterior angles. Interior angles are three angles within a triangle. Exterior angles are formed when the sides of a triangle are extended to infinity.

Therefore, exterior angles outside of a triangle are formed between one side of a triangle and the extended side. Every exterior angle is adjacent to an interior angle. Adjacent angles are angles that have a common vertex and side.

The figure below shows the angle of a triangle. The interior angles are a, b, and c, while the exterior angles are d, e, and f.

How do you find the angles of a triangle?

To find the angles of a triangle, you need to remember the following three properties of triangles:

Triangle angle sum theorem: This states that the sum of all three interior angles of a triangle equals 180 degrees.

a + b + c = 180º

Triangle Exterior Angle Theorem: This states that the exterior angle is equal to the sum of two opposite and nonadjacent interior angles.

f = b + a

e = c + b

d = b + c

straight angles. The angular measure on a straight line is equal to 180º

c + f = 180 degrees

a + d = 180 degrees

e + b = 180 degrees

Let’s solve a few sample problems.

example 1

Calculate the size of the missing angle x in the lower triangle.

solution

By the sum of the triangle angles, theorem, we have

x + 84º + 43º = 180º

Simplify.

x + 127º = 180º

Subtract 127º from both sides.

x + 127º – 127º = 180º – 127º

x = 53°

Therefore, the size of the missing angle is 53º.

example 2

Find the size of the interior angles of a triangle that form consecutive positive integers.

solution

Since a triangle has three interior angles, let the consecutive angles be:

⇒1. angle = x

⇒ 2nd angle = x + 1

⇒3. angle = x + 2

But we know that the sum of the three angles equals 180 degrees, so

⇒ x + x + 1 + x + 2 = 180°

⇒ 3x + 3 = 180°

⇒ 3x = 177°

x = 59°

Now substitute the value of x in the original three equations.

⇒1. Angle = x = 59°

⇒ 2nd angle = x + 1 = 59° + 1 = 60°

⇒3. Angle = x + 2 = 59° + 2 = 61°

So the consecutive interior angles of the triangle are: 59°, 60° and 61°.

Example 3

Find the interior angles of the triangle whose angles are given as; 2y°, (3y + 15)° and (2y + 25)°.

solution

In the triangle around the interior angle = 180°

2y° + (3y + 15)° + (2y + 25)° = 180°

Simplify.

2y + 3y + 2y + 15° + 25° = 180°

7y + 40° = 180°

Subtract 40° on both sides.

7y + 40° – 40° = 180° – 40°

7y = 140°

Divide both sides by 7.

y = 140/7

y = 20°

Substitute,

2y°= 2(20)° = 40°

(3y + 15)° = (3 x 20 + 15)° = 75°

(2y + 25)° = (2 x 20 + 25)° = 65°

The three interior angles of a triangle are 40°, 75° and 65°.

example 4

Find the value of the missing angles in the chart below.

solution

By the triangle exterior angle theorem we have;

(2x + 10)° = 63° + 87°

Simplify

2x + 10° = 150°

Deduct 10° on both sides.

2x + 10° – 10 = 150° – 10

2x = 140°

Divide both sides by 2 to get;

x = 70 degrees

Now by substitution;

(2x + 10)° = 2(70°) + 10° = 140° + 10° = 150°

The outer angle is therefore 150°

However, straight angles add up to 180°. So we have;

y + 150° = 180°

Subtract 150° on both sides.

y + 150° – 150° = 180° – 150°

y = 30°

Therefore the missing angles are 30° and 150°.

Example 5

The interior angles of a triangle are in the ratio 4:11:15. Find the angles.

solution

Let x be the common ratio of the three angles. So the angles are

4x, 11x and 15x.

In a triangle, the sum of the three angles = 180°

4x + 11x + 15x = 180°

Simplify.

30x = 180°

Divide 30 on both sides.

x = 180°/30

x = 6°

Substitute the value of x.

4x = 4(6)° = 24°

11x = 11(6)° = 66°

15x = 15(6)° = 90°

So the angles of the triangle are 24°, 66° and 90°.

Example 6

Find the size of the x and y angles in the diagram below.

solution

Exterior angle = Sum of two non-adjacent interior angles.

60° + 76° = x

x = 136 degrees

Likewise, the sum of the interior angles = 180°. Because of this,

60° + 76° + y = 180°

136° + y = 180°

Subtract 136° on both sides.

136° – 136° + y = 180° – 136

y = 44°

Therefore, the magnitude of the angles x and y are 136° and 44°, respectively.

Example 7

The three angles of a given triangle are such that the first angle is 20% less than the second angle and the third 20% greater than the second angle. Find the magnitude of the three angles.

solution

The second angle is x

First angle = x – 20x/100 = x – 0.2x

Third angle = x + 20x/100 = x + 0.2x

Sum of the three angles = 180 degrees.

x + x – 0.2x + x + 0.2x = 180°

Simplify.

3x = 180°

x = 60 degrees

Because of this,

2. second angle = 60°

1. Angle = 48°

3. Angle = 72°

So the three angles of a triangle are 60°, 48° and 72°.

example 8

Calculate the size of the angles p, q, r and s in the diagram below.

solution

Exterior angle = sum of the two non-adjacent interior angles.

140° = p + r …………. (I)

This is an isosceles triangle, so

q = r

Angle on a straight line = 180°

140° + q = 180°

subtract 140 from both sides to get

q = 40°

But q = r, so r is also 40°

r + s = 180° (linear angles)

40° + s = 180°

s = 140°

Sum of interior angles = 180°

p + q + r = 180°

p + 40° + 40° = 180°

p = 180° – 80°

p = 100°

Abc is a right triangle where A 90 and Ab Ac find B and C

ABC is a right triangle where A = 90° and AB = AC. Find B and C

Solution:

given,

ABC is a right triangle where ∠A = 90° and AB = AC.

Find:

We have to determine ∠B and ∠C

solution

In the right ∆ABC,

AB = AC

∴∠B = ∠C

∠A + ∠B + ∠C = 180°

90° + ∠B + ∠B = 180°

2∠B = 180° – 90°

2∠B = 90°

∠B = (90°)/2 = 45°

Answers:

∠B = 45° and ∠C = 45°

Alternative method:

From the given we can say that ∠B and ∠C are congruent since the corresponding angles of two equal sides are equal.

Also, the sum of two angles other than the right angle in a right triangle equals 90 degrees.

∠B + ∠C = 90°

∠B + ∠B = 90°

2∠B = 90°

∠B = (90°)/2 = 45°

Hence ∠B = ∠C = 45°.

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Pointed, blunt, isosceles, equilateral… When it comes to triangles, there are many different varieties, but few that are “special.” These special triangles have sides and angles that are consistent and predictable and can be used to shortcut your way through your geometry or trigonometry problems. And a 30-60-90 triangle – pronounced “thirty sixty ninety” – is actually a very special kind of triangle.

In this guide, we’ll explain what a 30-60-90 triangle is, why it works, and when (and how) to apply your knowledge of it. So let’s tackle it!

What is a 30-60-90 triangle?

A 30-60-90 triangle is a special right triangle (a right triangle is any triangle that contains a 90-degree angle) that always has angles of 30 degrees, 60 degrees, and 90 degrees. Since it is a special triangle, it also has side length values ​​that are always in a constant ratio to each other.

The basic triangle ratio 30-60-90 is:

Side opposite the 30° angle: $x$

Side opposite the 60° angle: $x * √3$

Side opposite the 90° angle: $2x$

For example, a 30-60-90 degree triangle might have the following side lengths:

2, 2√3, 4

7, 7√3, 14

√3, 3, 2√3 (Why is the longer leg 3? In this triangle, the shortest leg ($x$) is $√3$, so for the longer leg, $x√3 = √3 * √3 = √9 = 3$. And the hypotenuse is 2 times the shortest leg, or $2√3$)

Etc.

The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. The side opposite the 60° angle is the mean longitude, since 60 degrees is the mean angle in this triangle. Finally, the side opposite 90° is always the longest side (the hypotenuse), because 90 degrees is the longest angle.

Although it may look similar to other types of right triangles, a 30-60-90 triangle is special because you only need three pieces of information to find any other measure. As long as you know the value of two angle measures and a side length (it doesn’t matter which side), you know everything you need to know about your triangle.

For example, we can use the triangle formula 30-60-90 to fill in any remaining information gaps of the triangles below.

example 1

We can see that this is a right triangle in which the hypotenuse is twice the length of one of the legs. This means that this must be a 30-60-90 triangle and the smaller specified side is opposite the 30°. So the longer leg needs to face the 60° angle and measure $6 * √3$ or $6√3$.

example 2

We can see that this must be a 30-60-90 triangle because we can see that this is a right triangle with a given measure, 30°. The unmarked angle must then be 60°. Since 18 is the measure opposite the 60° angle, it must equal $x√3$. The shortest leg must then measure $18/√3$. (Note that the leg length is actually $18/{√3} * {√3}/{√3} = {18√3}/3 = 6√3$ since a denominator cannot contain a root/square root). And the hypotenuse will be $2(18/√3)$ (Note that again you can’t have a root in the denominator, so the final answer is really 2 times the leg length of $6√3$ => $12√3$ ).

Example 3

Again, we are given two angle measurements (90° and 60°), so the third measurement is 30°. Since this is a 30-60-90 triangle and the hypotenuse is 30, the shortest leg is 15 and the longer leg is 15√3.

No need to consult the Magic Eight Ball – these rules always work.

Why it works: 30-60-90 Triangle Theorem Proof

But why does this particular triangle work the way it does? How do we know these rules are legitimate? Let’s go through exactly how the 30-60-90 triangle theorem works and prove why these side lengths will always be consistent.

First, let’s forget right triangles for a second and look at an equilateral triangle.

An equilateral triangle is a triangle that has all equal sides and all equal angles. Since the interior angles of a triangle always add up to 180° and 180/3 = 60$, an equilateral triangle always has three 60° angles.

Now let’s drop an altitude down from the top angle to the base of the triangle.

We have now created two right angles and two congruent (equal) triangles.

How do we know they are equal triangles? Since we subtracted an altitude from an equilateral triangle, we split the base exactly in half. The new triangles also share a side length (the height) and they each have the same hypotenuse length. Since they have three side lengths in common (SSS), this means the triangles are congruent.

Note: The two triangles are not only congruent based on the principles of side-side-side-lengths, or SSS, but also based on side-angle-side (SAS), angle-angle-side (AAS), and angle measurements – Side Angle (ASA). Basically? They are definitely congruent.

Now that we have proved the congruence of the two new triangles, we can see that the apex angles must each equal 30 degrees (because each triangle already has angles of 90° and 60° and must add up to 180°). This means we made two 30-60-90 triangles.

And because we know that we’ve bisected the base of the equilateral triangle, we can see that the side opposite the 30° angle (the shortest side) of each of our 30-60-90 triangles is exactly half the length of the hypotenuse .

So let’s call our original side length $x$ and our halved length $x/2$.

Now all we have to do is find our median length that the two triangles have in common. To do this, we can simply use the Pythagorean theorem.

$a^2 + b^2 = c^2$

$(x/2)^2 + b^2 = x^2$

$b^2 = x^2 – ({x^2}/4)$

$b^2 = {4x^2}/4 – {x^2}/4$

$b^2 = {3x^2}/4$

$b = {√3x}/2$

So we are left with: $x/2, {x√3}/2, x$

Now let’s multiply each bar by 2 just to make life easier and avoid all the breaks. So we are left with:

$x$, $x√3$, $2x$

We can therefore see that a 30-60-90 triangle will always have consistent side lengths of $x$, $x√3$ and $2x$ (or $x/2$, ${√3x}/2) $ and $x$).

Fortunately, we can prove the 30-60-90 triangle rules without all of that.

When to use 30-60-90 triangle rules?

Knowing the 30-60-90 triangle rules can save you time and energy on a variety of different math problems, especially a variety of geometry and trigonometry problems.

geometry

Proper understanding of the 30-60-90 triangles will allow you to solve geometry questions that would either be impossible to solve without knowledge of these ratio rules, or at the very least would require a great deal of time and effort to solve the “long way”.

With special triangle ratios, you can find missing triangle heights or leg lengths (without having to use the Pythagorean Theorem), find the area of ​​a triangle using missing height or base length information, and quickly calculate perimeters.

Anytime you need speed to answer a question, it comes in handy to remember shortcuts like your 30-60-90 rules.

trigonometry

Memorizing and understanding the triangle ratio 30-60-90 will also allow you to solve many trigonometric problems without needing a calculator or having to approximate your answers in decimal form.

A 30-60-90 triangle has fairly simple sines, cosines, and tangents for each angle (and these measurements will always be consistent).

The sine of 30° is always $1/2$.

The cosine of 60° is always $1/2$.

Although the other sine, cosine, and tangent are fairly easy, these two are the easiest to remember and are likely to show up in tests. So if you know these rules, you can find these trigonometric measurements as quickly as possible.

Tips for remembering the 30-60-90 rules

You know these 30-60-90 ratio rules are useful, but how do you keep the information in your head? Remembering the 30-60-90 triangle rules means remembering the ratio of 1:√3:2 and knowing that the shortest side always opposes the shortest angle (30°) and the longest side always opposes lies at the greatest angle (90°).

Some people remember the ratio by thinking “$\bi x$, $\bo 2 \bi x$, $\bi x \bo √ \bo3$” because the “1, 2, 3” sequence is usually easy to understand remember. The only caveat when using this technique is to remember that the longest side is actually $2x$, not $x$ times $√3$.

Another way to remember your ratios is to use a mnemonic pun for the ratio 1:root 3:2 in order. For example, “Jackie Mitchell beat Lou Gehrig and ‘won Ruthy too'”: one root three two. (And it’s a true fact of baseball history!)

Play around with your own mnemonics if they don’t appeal to you – sing the ratio to a song, come up with your own “one, square root, three, two” phrases, or come up with a ratio poem. You can also just remember that a 30-60-90 triangle is half an equilateral triangle and figure out the measurements from there if you’re not keen on memorizing them.

However, it is useful to remember these 30-60-90 rules and keep these ratios in mind for your future geometry and trigonometry questions.

Memorization is your friend, but you can make it happen.

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Example 30-60-90 questions

Now that we’ve looked at the hows and whys of 30-60-90 triangles, let’s work through some practice problems.

geometry

A construction worker leans a 40-foot ladder against the side of a building at a 30-degree angle from the ground. The ground is level and the side of the building is perpendicular to the ground. How far does the ladder reach up the building to the next foot?

Without knowing our specific 30-60-90 triangle rules, we would have to use trigonometry and a calculator to find the solution to this problem since we only have one side measurement of a triangle. But because we know this is a special triangle, we can find the answer in seconds.

If the building and the ground are perpendicular to each other, the building and the ground must form a right angle (90°). It goes without saying that the ladder hits the ground at an angle of 30°. So we can see that the remaining angle must be 60°, making this a 30-60-90 triangle.

Now we know that the hypotenuse (longest side) of this 30-60-90 is 40 feet long, which means the shortest side is half that length. (Remember that the longest side is always twice the length—$2x$—of the shortest side.) Because the shortest side is opposite the 30° angle, and that angle is the ladder’s degree from the ground, that means Die Top of the ladder meets the building 20 feet off the ground.

Our final answer is 20 feet.

trigonometry

If in a right triangle sin Θ = $1/2$ and the shortest leg length is 8. What is the length of the missing side that is NOT the hypotenuse?

Now that you know your 30-60-90 rules, you can solve this problem without needing the Pythagorean theorem or a calculator.

We’ve been told that this is a right triangle, and we know from our special right triangle rules that sine is 30° = $1/2$. So the missing angle must be 60 degrees, making a 30-60-90 triangle.

And because this is a 30-60-90 triangle and we’re told the shortest side is 8, the hypotenuse has to be 16 and the missing side has to be $8 * √3$ or $8√3$.

Our final answer is 8√3.

The take aways

If you remember the rules for 30-60-90 triangles, you can shortcut your way through a variety of math problems. However, remember that you can solve most problems without them, although knowing these rules is a handy tool to keep in your belt.

Keep an eye on the rules of $x$, $x√3$, $2x$, and 30-60-90 for whatever makes sense to you and try to keep them as clear as possible, but don’t guess panicking fades out when it’s crunch time. Either way, you got this.

And if you need more practice, check out this 30-60-90 triangle quiz. Have fun testing!