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 180o. In the image above m \angle{ABD} = 135o, and m \angle{CBD} = 45o. 135o + 45o = 180o.

Angle Measurement Postulate


'The measure of an angle is between 0 and 180o.'

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 135o, what could be the measurement of the angles of their supplements?. Since the sum of the supplementary angles is equal to 180o, We get the difference between 180o and 135o to get the measurement of the angle of their supplements. So, if we want to get m \angle{b} ,we will have 180o - 135o. Thus m \angle{b} = 45o. For m \angle{c} ,since m \angle{d} is equal to m \angle{a} , therefore m \angle{c} is also equal to 45o. 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 45o, what could be the measurement of the angles of their complements?. Since the sum of the complementary angles is equal to 90o, We get the difference between90o and 45o to get the measurement of the angle of their complements. So if we want to get m \angle{2} , we will have 90o - 45o. Thus m \angle{2} = 45o. For m \angle{3} Since m \angle{4} is equal to m \angle{1} , therefore m \angle{c} is also equal to 45o. 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 90o. 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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