**Angle relationships in parallel lines and triangles :**

In this section, we are going to see the angle relationships, when two parallel lines are cut by a transversal and in triangles.

**Corresponding angles : **

Angles lie on the same side of the transversal t, on the same side of lines a and b.

Example : ∠ 1 and ∠ 5

**Alternate interior angles :**

Angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 6

**Alternate exterior angles :**

Angles lie on opposite sides of the transversal t, outside lines a and b.

Example : ∠ 1 and ∠ 8

**Same-side interior angles :**

Angles lie on the same side of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 5

In this section, we are going to see the angle relationships in triangles through the following steps.

**Step 1 : **

Draw a triangle and cut it out. Label the angles A, B, and C.

**Step 2 : **

Tear off each “corner” of the triangle. Each corner includes the vertex of one angle of the triangle.

**Step 3 : **

Arrange the vertices of the triangle around a point so that none of your corners overlap and there are no gaps between them.

**Step 4 : **

What do you notice about how the angles fit together around a point ?

The angles form a straight angle.

**Step 5 : **

What do you notice about how the angles fit together around a point ?

180°

**Step 6 : **

Describe the relationship among the measures of the angles of triangle ABC ?

The sum of the angle measures is 180°.

**Step 7 : **

What does the triangle sum theorem state ?

The triangle sum theorem states that for triangle ABC,

m∠A + m∠B + m∠C = 180°

The theorem states that the measure of an** exterior** angle is equal to the sum of its **remote interior** angles.

That is,

m∠1 + m∠2 = m∠4

**Problem 1 :**

In the figure given below, let the lines l₁ and l₂ be parallel and m is transversal. If ∠F = 65°, find the measure of each of the remaining angles.

**Solution : **

From the given figure,

∠F and ∠H are vertically opposite angles and they are equal.

Then, ∠H = ∠F -------> ∠H = 65°

∠H and ∠D are corresponding angles and they are equal.

Then, ∠D = ∠H -------> ∠D = 65°

∠D and ∠B are vertically opposite angles and they are equal.

Then, ∠B = ∠D -------> ∠B = 65°

∠F and ∠E are together form a straight angle.

Then, we have

∠F + ∠E = 180°

Plug ∠F = 65°

∠F + ∠E = 180°

65° + ∠E = 180°

∠E = 115°

∠E and ∠G are vertically opposite angles and they are equal.

Then, ∠G = ∠E -------> ∠G = 115°

∠G and ∠C are corresponding angles and they are equal.

Then, ∠C = ∠G -------> ∠C = 115°

∠C and ∠A are vertically opposite angles and they are equal.

Then, ∠A = ∠C -------> ∠A = 115°

Therefore,

∠A = ∠C = ∠E = ∠G = 115°

∠B = ∠D = ∠F = ∠H = 65°

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

**Problem 2 : **

Can 30°, 60° and 90° be the angles of a triangle ?

**Solution :**

Let us add all the three given angles and check whether the sum is equal to 180°.

30° + 60° + 90° = 180°

Since the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

**Problem 3 :**

In a triangle, if the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

**Solution :**

Let "x" be the first angle.

The second angle = x + 5

The third angle = x + 5 + 5 = x + 10

We know that,

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

x + (x+5) + (x+10) = 180°

3x + 15 = 180

3x = 165

x = 55

The first angle = 55°

The second angle = 55 + 5 = 60°

The third angle = 60 + 5 = 65°

Hence, the three angles of a triangle are 55°, 60° and 65°.

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

**Problem 4 :**

Find m∠W and m∠X in the triangle given below.

**Solution : **

**Step 1 : **

Write the Exterior Angle Theorem as it applies to this triangle.

m∠W + m∠X = m∠WYZ

**Step 2 : **

Substitute the given angle measures.

(4y - 4)° + 3y° = 52°

**Step 3 : **

Solve the equation for y.

(4y - 4)° + 3y° = 52°

4y - 4 + 3y = 52

Combine the like terms.

7y - 4 = 52

Add 4 to both sides.

7y - 4 + 4 = 52 + 4

Simplify.

7y = 56

Divide both sides by 7.

7y / 7 = 56 / 7

y = 8

**Step 4 : **

Use the value of y to find m∠W and m∠X.

m∠W = 4y - 4

m∠W = 4(8) - 4

m∠W = 28

m∠X = 3y

m∠X = 3(8)

m∠X = 24

So, m∠W = 28° and m∠X = 24°.

After having gone through the stuff given above, we hope that the students would have understood "Angle relationships in parallel lines and triangles".

Apart from the stuff given above, if you want to know more about "Angle relationships in parallel lines and triangles", please click here

Apart from the stuff given on "Angle relationships in parallel lines and triangles", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**