Subject: Geometry

# Angles

### Postulates and Theorems

A postulate is a statement that is assumed to be true without any proof. However Theorem is a statement derived from definitions and proofs.

### Angle postulates

**Angle Addition Postulate**

if point B is in the interior of \angle{ADC} then, m \angle{ADC} = m \angle{ADB} + m \angle{BDC}

**Supplement Postulate**

**Two angles aresupplementary if their angles form a linear pair. m \angle{ABD} and m \angle{CBD} are supplementary angles.**

The sum of supplementary angles is equal to 180^{o}. In the image above m \angle{ABD} = 135^{o}, and m \angle{CBD} = 45^{o}. 135^{o} + 45^{o} = 180^{o}.

**Angle Measurement Postulate**

*'The measure of an angle is between 0 and 180*'

^{o}.### Angle Theorems

**Vertical angles are congruent. (Vertical Angle Theorem)**

**\angle{1} and \angle{2} are vertical angles. \angle{3} and \angle{4} are also vertical angles. Thus m \angle{1} = m \angle{2} ; m \angle{3} = m \angle{4}**

**Supplements of congruent angles are equal (Supplement Theorem)**

Suppose \angle{a} and \angle{d} are congruent angles. Their supplements are \angle{b} and \angle{c} respectively. If m \angle{a} and \angle{d} is equal to 135^{o}, what could be the measurement of the angles of their supplements?. Since the sum of the supplementary angles is equal to 180^{o}, We get the difference between 180^{o} and 135^{o} to get the measurement of the angle of their supplements. So, if we want to get m \angle{b} ,we will have 180^{o} - 135^{o}. Thus m \angle{b} = 45^{o}. For m \angle{c} ,since m \angle{d} is equal to m \angle{a} , therefore m \angle{c} is also equal to 45^{o}. **Thus the supplements of congruent angles are equal.**

**If \angle{a} = \angle{d} , then \angle{b} = \angle{c}**

**Complements of congruent angles are equal.**

Suppose \angle{1} and \angle{4} are congruent angles. Their complements are \angle{2} and \angle{3} respectively. If m \angle{1} and \angle{4} is equal to 45^{o}, what could be the measurement of the angles of their complements?. Since the sum of the complementary angles is equal to 90^{o}, We get the difference between90^{o} and 45^{o} to get the measurement of the angle of their complements. So if we want to get m \angle{2} , we will have 90^{o} - 45^{o}. Thus m \angle{2} = 45^{o}. For m \angle{3} Since m \angle{4} is equal to m \angle{1} , therefore m \angle{c} is also equal to 45^{o}. **Thus the complements of congruent angles are equal.**

\angle{1} \cong \angle{4} ; \angle{2} \cong \angle{3}

**If two angles are complementary, then both are acute angles**

**If two angles are right angles, then these angles are congruent**

Right angles have fixed measurement of angles which is 90^{o}. Thus if two angles are both right angles then, they are congruent.

**If two angles are congruent and are supplementary, then each is a right angle**.

**If two lines which are perpendicular form one right angle, then it also form four right angles.**

**Try to answer the following:**

In the image below:

- Name the supplementary angles and give their angle measurements
- Using the angle addition postulate, write an equation using the angles on the image

1. Supplementary angles:

a. \angle{ABD}=50^o; \angle{CBE}=180^o-50^o=130^o

b. \angle{ABE}=50^o+90^o=140^o; \angle{CBE}=180^o-40^o=40^o

2. Angle Addition Postulate:

a. m\angle{ABC}=m\angle{ABD}+m\angle{ABE}

b. m\angle{ABE}=m\angle{ABD}+m\angle{BDE}

c. m\angle{DBC}=m\angle{DBE}+m\angle{EBC}

In the image below:

- identify all the supplementary angles
- identify all complimentary angles
- identify adjacent angles
- Using the angle addition postulate, write an equation using the angles on the image

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