Angles related to Intersecting Lines

 
8.2  Angles related to Intersecting Lines
 
Example

(i) Vertically opposite angles at intersecting lines are equal.

\(\begin{aligned}∠a &= ∠c\\\\ ∠b &= ∠d \end{aligned}\)

(ii) The sum of adjacent angles at intersecting lines is \(180^{\circ}\).

\(\begin{aligned}∠a+\angle d &= 180^\circ\\\\ ∠d+\angle c&= 180^\circ\\\\ ∠c+\angle b&= 180^\circ\\\\ ∠b+\angle a&= 180^\circ \end{aligned}\)

(iii) If two intersecting lines are perpendicular to each other, then all angles at the intersecting lines have the same size of \(90^{\circ}\).

 
Determine the values of the angles at intersecting lines:
 
Example

The following diagram shows \(PSQ\), \(RSTU\) and \(PTV\) are straight lines.

Calculate the values of \(x\)\(\) and \( y\).

Noted that the sum of adjacent angles at intersecting lines is \(180^{\circ}\).

So,

\(\begin{aligned}x + 135^{\circ} &= 180^{\circ} \\\\x&=180^\circ-135^\circ \\\\x&=55^\circ. \end{aligned}\)

Also, the vertically opposite angles at intersecting lines are equal.

Thus, \(y = 62^{\circ}\).

 

 

Angles related to Intersecting Lines

 
8.2  Angles related to Intersecting Lines
 
Example

(i) Vertically opposite angles at intersecting lines are equal.

\(\begin{aligned}∠a &= ∠c\\\\ ∠b &= ∠d \end{aligned}\)

(ii) The sum of adjacent angles at intersecting lines is \(180^{\circ}\).

\(\begin{aligned}∠a+\angle d &= 180^\circ\\\\ ∠d+\angle c&= 180^\circ\\\\ ∠c+\angle b&= 180^\circ\\\\ ∠b+\angle a&= 180^\circ \end{aligned}\)

(iii) If two intersecting lines are perpendicular to each other, then all angles at the intersecting lines have the same size of \(90^{\circ}\).

 
Determine the values of the angles at intersecting lines:
 
Example

The following diagram shows \(PSQ\), \(RSTU\) and \(PTV\) are straight lines.

Calculate the values of \(x\)\(\) and \( y\).

Noted that the sum of adjacent angles at intersecting lines is \(180^{\circ}\).

So,

\(\begin{aligned}x + 135^{\circ} &= 180^{\circ} \\\\x&=180^\circ-135^\circ \\\\x&=55^\circ. \end{aligned}\)

Also, the vertically opposite angles at intersecting lines are equal.

Thus, \(y = 62^{\circ}\).