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In this chapter, friend will check out the relationships in between pairs of angle that are produced when directly lines intersect (meet or cross). You will examine the bag of angles that are created by perpendicular lines, by any kind of two intersecting lines, and by a 3rd line the cuts 2 parallel lines. Friend will pertained to understand what is intended by vertically the opposite angles, corresponding angles, alternating angles and co-interior angles. Girlfriend will be able to identify various angle pairs, and also then use your knowledge to aid you work-related out unknown angle in geometric figures.

Angles on a directly line

Sum of angle on a right line

In the figures below, each angle is provided a brand from 1 come 5.

usage a protractor to measure the size of all the angles in each figure. Create your answer on every figure.





usage your answers to fill in the angle size below. \( \hat1 + \hat2 = \text______^\circ \) \( \hat3 + \hat4 + \hat5= \text______^\circ \)

The sum of angle that are formed on a directly line is equal to 180°. (We can shorten this residential property as: \(\angle\)s on a straight line.)

two angles who sizes include up to 180° are additionally called supplementary angles, for example \( \hat1 + \hat2\).

angle that re-superstructure a vertex and a common side are claimed to it is in adjacent. Therefore \( \hat1 + \hat2\) room therefore also called supplementary surrounding angles.

When 2 lines room perpendicular, their nearby supplementary angles room each same to 90°.

In the drawing below, DC A and DC B are adjacent supplementary angles due to the fact that they are next to each other (adjacent) and they add up to 180° (supplementary).


Finding unknown angle on directly lines

Work the end the size of the unknown angle below. Build an equation each time together you solve these geometric problems. Constantly give a reason for every statement girlfriend make.

calculate the dimension of \(a\).


\( \beginalign a + 63^\circ &= \text______ <\angle\texts ~ above a directly line> \\ a &= \text______ - 63^\circ \\ &= \text______ \endalign\)

calculation the dimension of \(x\).


calculate the dimension of \(y\).


Finding an ext unknown angles on directly lines

calculation the size of:


\(x\) \(\hatECB\) calculation the size of:


\(m\) \(\hatSQR\) calculate the dimension of:


\(x\) \(\hatHEF\) calculation the size of:


\(k\) \(\hatTYP\) calculation the dimension of:


\(p\) \(\hatJKR\)

Vertically the contrary angles

What room vertically opposite angles?

use a protractor to measure the size of all the angles in the figure. Compose your answers on the figure.


notice which angles space equal and also how these same angles room formed.

Vertically the opposite angles (vert. Opp. \(\angle\)s) are the angle opposite each other when two present intersect.

Vertically the contrary angles room always equal.


Finding unknown angles

Calculate the size of the unknown angles in the following figures. Constantly give a factor for every statement friend make.

calculate \(x,~ y\) and \(z\).


\( \beginalign x &= \text______^\circ &&<\textvert. Opp.\angle\texts> \\ \\ y + 105^\circ &= \text______^\circ &&<\angle\texts on a directly line> \\ y &= \text______ - 105^\circ && \\ & = \text______ \\ \\ z &= \text______ &&<\textvert. Opp.\angle\texts> \endalign\)

calculation \(j,~ k\) and \(l\).


calculate \(a,~ b,~ c\) and also \(d\).


Equations making use of vertically the contrary angles

Vertically opposite angles are always equal. We deserve to use this residential property to build an equation. Climate we solve the equation to uncover the value of the unknown variable.

calculate the value of \(m\).


\( \beginalign m + 20^\circ &= 100^\circ <\textvert. Opp.\angle\texts> \\ m &= 100^\circ - 20^\circ \\ &= \text______ \endalign\) calculate the value of \(t\).


calculate the worth of \(p\).


calculation the worth of \(z\).


calculation the value of \(y\).


calculate the value of \(r\).


Lines intersected through a transversal

Pairs the angles created by a transversal

A transversal is a heat that the cross at the very least two other lines.


When a transversal intersects two lines, we have the right to compare the sets of angle on the 2 lines by feather at your positions.

The angle that lie on the exact same side that the transversal and also are in corresponding positions are called corresponding angles (corr.\(\angle\)s). In the figure, this are matching angles:


\(a\) and also \(e\) \(b\) and \(f\) \( d\) and \(h\) \(c\) and \(g\). In the figure, \(a\) and also \(e\) room both left the the transversal and over a line.

write down the place of the following corresponding angles. The an initial one is done for you.

\(b\) and \(f\): ideal of the transversal and over lines

\(d\) and \(h\):

\(c\) and \(g\):

Alternate angles (alt.\(\angle\)s) lie on opposite political parties of the transversal, but are not adjacent or vertically opposite. Once the alternating angles lie between the two lines, lock are referred to as alternate inner angles. In the figure, these are alternate interior angles:

\(d\) and \(f\) \(c\) and \(e\)

When the alternate angles lie outside of the two lines, lock are referred to as alternate exterior angles. In the figure, this are alternating exterior angles:

\(a\) and also \(g\) \(b\) and \(h\) create down the ar of the following alternating angles:

\(d\) and also \(f\):

\(c\) and \(e\):

\(a\) and also \(g\):

\(b\) and also \(h\):

Co-interior angles (co-int.\(\angle\)s) lie on the same side the the transversal and also between the 2 lines. In the figure, these space co-interior angles:

\(c\) and \(f\) \(d\) and also \(e\) compose down the ar of the complying with co-interior angles:

\(d\) and also \(e\):

\(c\) and \(f\):

Identifying varieties of angles

Two lines space intersected by a transversal as presented below.


Write down the complying with pairs of angles:

two pairs of equivalent angles: 2 pairs of alternate interior angles: two pairs of alternative exterior angles: two pairs of co-interior angles: two pairs the vertically the opposite angles:

Parallel present intersected by a transversal

Investigating angle sizes

In the figure below left, EF is a transversal to abdominal and CD. In the figure listed below right, PQ is a transversal to parallel lines JK and LM.


usage a protractor to measure the sizes of every the angles in each figure. Write the dimensions on the figures. use your measurements to finish the complying with table.


When two lines space not parallel

When two lines are parallel


\( \hat1 = \text_______;~\hat5 = \text_______\)

\( \hat4 = \text_______;~\hat8 = \text_______\)

\( \hat2 = \text_______;~\hat4 = \text_______\)

\( \hat3 = \text_______;~\hat7 = \text_______\)

\( \hat9 = \text_______;~\hat13 = \text_______\)

\( \hat12 = \text_______;~\hat16 = \text_______\)

\( \hat10 = \text_______;~\hat14 = \text_______\)

\( \hat11 = \text_______;~\hat15 = \text_______\)


\( \hat4 = \text_______;~\hat6 = \text_______\)

\( \hat3 = \text_______;~\hat5 = \text_______\)

\( \hat12 = \text_______;~\hat14 = \text_______\)

\( \hat11 = \text_______;~\hat13 = \text_______\)


\( \hat1 = \text_______;~\hat7 = \text_______\)

\( \hat2 = \text_______;~\hat8 = \text_______\)

\( \hat9 = \text_______;~\hat15 = \text_______\)

\( \hat10 = \text_______;~\hat16 = \text_______\)


\( \hat4 + \hat5 = \text_______\)

\( \hat3 + \hat6 = \text_______\)

\( \hat12 + \hat13 = \text_______\)

\( \hat11 + \hat14 = \text_______\)

look at at your completed table in concern 2. What execute you an alert about the angles formed when a transversal intersects parallel lines?

When lines room parallel:

equivalent angles space equal alternate interior angles space equal alternate exterior angles space equal co-interior angles include up to 180°

Identifying angles on parallel lines

to fill in the corresponding angles come those given.


to fill in the alternate exterior angles.


fill in the alternative interior angles. one the two pairs the co-interior angles in every figure.


there is no measuring, to fill in all the angle in the following numbers that are equal to \(x\) and also \(y\). describe your reasons for each \(x\) and \(y\) that you to fill in to your partner.

Finding unknown angle on parallel lines

Working out unknown angles

Work the end the sizes of the unknown angles. Offer reasons for your answers. (The first one has actually been done together an example.)

find the size of \(x,~y\) and also \(z\).


\( \beginalign x &= 74^\circ &&<\textalt.\angle\text with provided 74^\circ; abdominal muscle \parallel CD> \\ \\ y &= 74^\circ &&<\textcorr.\angle\text v x; ab \parallel CD> \\ \textor y &= 74^\circ &&<\textvert. Opp.\angle\text with offered 74^\circ> \\ \\ z &= 106^\circ &&<\textco-int.\angle\text with x; abdominal muscle \parallel CD> \\ \textor z &= 106^\circ &&<\angle\texts on a directly line> \endalign\)

job-related out the size of \(p,~ q\) and also \(r\).


uncover the size of \(a,~b,~c\) and also \(d\).


find the size of every the angles in this figure.


discover the size of all the angles. (Can girlfriend see 2 transversals and two to adjust of parallel lines?)



Two angle in the adhering to diagram are provided as \(x\) and also \(y\). Fill in all the angle that room equal to \(x\) and \(y\).


Sum that the angle in a quadrilateral

The diagram below is a section of the ahead diagram.

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What sort of quadrilateral is in the diagram? offer a factor for your answer. Look in ~ the peak left intersection. Finish the complying with equation:

Angles roughly a point\( = 360^\circ\)

\(\therefore x + y+ \text______ + \text______ = 360^\circ\)

Look at the inner angles of the quadrilateral. Complete the following equations:

Sum of angle in the quadrilateral \(= x + y + + \text______ + \text______\)

From question 2: \(x + y+ \text______ + \text______ = 360^\circ\)

\(\therefore\) amount of angles in a square = \(\text______^\circ\)

Can you think that another way to usage the diagram above to job-related out the sum of the angle in a quadrilateral?

Solving more geometric problems

Angle relationship on parallel lines

calculation the sizes of \(\hat1\) to \(\hat7\).


calculation the size of \(x,~y\) and \(z\).


calculate the sizes of \(a, ~b, ~c\) and \(d\).


calculate the size of \(x\).


calculate the dimension of \(x\).


calculation the dimension of \(x\).


calculation the sizes of \(a\) and \(\hatCEP\).


Including properties of triangles and also quadrilaterals

calculate the sizes of \(\hat1\) to \(\hat6\).


RSTU is a trapezium. Calculation the sizes of \(\hatT\) and also \(\hatR\).


JKLM is a rhombus. Calculation the sizes of \(\hatJML, \hatM_2\) and also \(\hatK_1\).


ABCD is a parallelogram. Calculation the size of \(\hatADB, \hatABD, \hatC\) and also \(\hatDBC\)


Look at the illustration below. Name the items listed alongside.


a pair of vertically opposite angles a pair of equivalent angles a pair of alternating interior angles a pair of co-interior angles In the diagram, ab \(\parallel\) CD. Calculation the size of \(\hatFHG, \hatF, \hatC\) and also \(\hatD\). Offer reasons for your answers.


In the diagram, yes sir = ON, KN \(\parallel\) LM, KL\(\parallel\) MN and \(\hatLKO = 160^\circ\).

Calculate the worth of \(x\). Offer reasons for her answers.