# Fundamental Geometric Concepts

**Angle-**a figure formed by two rays with same initial point. The initial point is its vertex and the two rays are its arms or sides.

**Type of angle:**

*Acute angle*-an angle whose measure lies between 0° and 90°.

*Right angle*-an angle whose measure is 90°.

*Obtuse angle*-an angle whose measure lies between 90° and 180°.

*Straight angle*-an angle whose measure is 180°.

*Reflex angle*-an angle whose measure lies between 180° and 360°.**Types of angles:**

*Adjacent angles*-two angles that have a common vertex, a common arm and their other arms lie on either side of the common arm.

*Linear pair*-two adjacent angles whose exterior arms are in a straight line. Thus if two adjacent angles form a linear pair, then their sum is 180°.

*Complementary angles*-two angles whose sum of measures is 90°.

*Supplementary angles*-two angles whose sum of measures is 180°.- Sum of angles at a point = 360°.
- Sum of angles at a point on a straight line = 180°.
- If the sum of two adjacent angles is 180°, then their exterior arms are in a straight line.
- If two straight lines intersect, then vertically opposite angles are equal.
*Transversal*-a line that interests two (or more) lines in a plane at distinct points.**Properties of angles associated with parallel lines**If a transversal meets two parallel lines, then:

corresponding angles are equal.

alternate angles are equal.

co-interior angles are supplementary.**Conditions of parallelism**If two lines are cut by a transversal such that a pair of:

corresponding angles is equal, then the lines are parallel.

alternate angles is equal, then the lines are parallel.

co-interior angles is supplementary, then the lines are parallel.- If two lines are parallel to a third line, then the lines are themselves parallel.

## Exercise

- Two angles are complementary and one angle is 10° less than three times the other, find the angles.
- Two supplementary angles are in the ratio 2:7, find the complement of the smaller angle.
- From the following diagram, find the value of x and hence complete the following:

(i) AOC =...

(ii) DOE =...

- In the figure below, AB and CD are straight lines. Find

(i) x, y and z if x = p and q = 80°

(ii) p if y : z = 2 : 3

(iii) z if p : q : x = 2 : 3 : 1

(iv) x if y = 40° and q = 2p +10°.

- Find the measure of each lettered angle in the following figures :

(i) (ii)

(iii) - Find the value of x from the following sketches :

(i) (ii) - In the following figure, find the value of x so that the lines AB
and CD may be parallel.

## Answers

**1.**25°, 65°

**2.**40°, 140°, 50°

**3.**35° (i) 125° (ii) 109°

**4.**(i) x = 50°, y = 50°, z = 130° (ii) 72° (iii) 120° (iv) 50°

**5.**(i) x = 42°, y = 63°, z = 75°, p = 138°

(ii) x = 75°, y = 15° (iii) x = 145°,y = 55°, z = 125°

**6.**(i) 38 (ii) 23

**7.**37°