C5.5; E5.5 - Angles at points and in triangles

C5.5; E5.5 - Angles at points and in triangles

Two straight lines cross, and one angle is 128∘128^\circ. Can you find the other three without measuring? C5.5 and E5.5 give the angle facts for points, straight lines, triangles and quadrilaterals, together with the notation and reasons that make a calculation convincing.

Angles at a point

An angle measures a turn. One complete turn returns to the starting ray; a half-turn points along the same straight line in the opposite direction.

Angles at a point and on a line

Angles filling one complete turn around a point sum to 360∘360^\circ. Adjacent angles filling a straight angle sum to 180∘180^\circ. When two straight lines cross, the angles directly opposite each other are vertically opposite angles, and they are equal.

“Vertically” does not mean up and down on the page. It identifies the opposite pair made by the same two intersecting lines. Adjacent angles share a ray; vertically opposite angles do not.

Why are the opposite angles equal? If one angle is aa and its neighbour is bb, then a+b=180∘a+b=180^\circ. If the next angle is cc, then b+c=180∘b+c=180^\circ too. Both aa and cc equal 180∘−b180^\circ-b, so a=ca=c.

[DIAGRAM: asset_name: Straight, opposite and full-turn angle facts; asset_slug: point_angle_facts; caption: Adjacent angles on a straight line sum to 180 degrees, opposite angles are equal, and a complete turn sums to 360 degrees.; description: Two exact panels show crossing lines with angles 128 and 52 degrees, and four angles around a point measuring 88, 115, 92 and 65 degrees. Arcs select non-overlapping regions and the straight lines genuinely continue through the vertex.]
Adjacent angles on a straight line sum to 180 degrees, opposite angles are equal, and a complete turn sums to 360 degrees.

Worked example 1: crossing straight lines

Question. Two straight lines intersect. One angle is 128∘128^\circ. Find each of the other three angles.

Plan. Find its adjacent angle using a straight line, then use vertically opposite equality.

adjacent angle=180∘−128∘=52∘angles on a straight lineopposite the 128∘ angle=128∘vertically opposite anglesopposite the 52∘ angle=52∘vertically opposite angles\begin{aligned} \text{adjacent angle}&=180^\circ-128^\circ=52^\circ &&\text{angles on a straight line}\\ \text{opposite the }128^\circ\text{ angle}&=128^\circ &&\text{vertically opposite angles}\\ \text{opposite the }52^\circ\text{ angle}&=52^\circ &&\text{vertically opposite angles} \end{aligned}

Answer. The other angles are 52∘, 128∘, 52∘52^\circ,\ 128^\circ,\ 52^\circ.

Check. 128+52+128+52=360128+52+128+52=360, and each adjacent pair sums to 180∘180^\circ.

Now the rays do not form two crossing straight lines. Use a complete turn rather than assuming any opposite-looking angles are equal.

Worked example 2: a full turn

Question. Four consecutive, non-overlapping angles around a point are 88∘, 115∘, 92∘88^\circ,\ 115^\circ,\ 92^\circ and x∘x^\circ. Find xx.

Plan. Subtract the three known angles from a complete turn.

88+115+92=295known part of the turnx=360−295angles around a point=65\begin{aligned} 88+115+92&=295 &&\text{known part of the turn}\\ x&=360-295 &&\text{angles around a point}\\ &=65 \end{aligned}

Answer. x=65x=65, so the missing angle is 65∘65^\circ.

Check. 88+115+92+65=36088+115+92+65=360. The four regions fill exactly one complete turn.

A pair is supplementary when its angles sum to 180∘180^\circ. The word describes the sum; “angles on a straight line” explains why that sum applies to a particular diagram.

Watch out

Vertically opposite angles require two straight intersecting lines. Four arbitrary rays do not create that equality.

Use only the complete-turn information given below.

Triangle and quadrilateral sums

For a closed shape, the interior angles belong to its vertices. Count them before choosing the total: a triangle has three, while a quadrilateral has four.

Triangle and quadrilateral angle sums

The interior angles of a triangle sum to 180∘180^\circ. The interior angles of a quadrilateral sum to 360∘360^\circ.

A diagonal divides a quadrilateral into two triangles, so its four interior angles together make 2×180∘=360∘2\times180^\circ=360^\circ. The rule does not require a rectangle or any equal sides. For a concave quadrilateral, include its reflex interior angle in the same total.

Equal-side information can provide extra angles: an isosceles triangle has equal angles opposite its equal sides. If its unequal vertex angle is 40∘40^\circ, its other two angles share 180∘−40∘=140∘180^\circ-40^\circ=140^\circ, giving 70∘70^\circ each.

Worked example 3: a triangle

Question. A triangle has interior angles 53∘, 68∘53^\circ,\ 68^\circ and t∘t^\circ. Find tt.

Plan. Subtract the known angles from the triangle total.

53+68=121known anglest=180−121angles in a triangle=59\begin{aligned} 53+68&=121 &&\text{known angles}\\ t&=180-121 &&\text{angles in a triangle}\\ &=59 \end{aligned}

Answer. t=59t=59.

Check. 53+68+59=18053+68+59=180. All three angles are positive and less than 180∘180^\circ, as required for a triangle.

Changing from three sides to four changes the total. Do not carry the 180∘180^\circ total into the next example.

Worked example 4: a quadrilateral

Question. Three interior angles of a quadrilateral are 86∘, 104∘86^\circ,\ 104^\circ and 97∘97^\circ. Find the fourth angle.

Plan. Use the quadrilateral total of 360∘360^\circ.

86+104+97=287three known anglesfourth angle=360∘−287∘angles in a quadrilateral=73∘\begin{aligned} 86+104+97&=287 &&\text{three known angles}\\ \text{fourth angle}&=360^\circ-287^\circ &&\text{angles in a quadrilateral}\\ &=73^\circ \end{aligned}

Answer. 73∘73^\circ.

Check. 86+104+97+73=36086+104+97+73=360. A rough estimate of 90+100+10090+100+100 leaves about 70∘70^\circ, matching the result.

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