This is the marking copy and it carries every answer. Her sheet is the other file — same numbering, same margin codes, so you can mark straight down the page.
Using the same diagram (ABEF a square, BCDE a rectangle, with A, B, C on one straight line and F, E, D on another), ∠ACF = 35° and ∠DCE = 18°. Find ∠FCE.
A B C
+--------------------+---------+
\ \35 /|
\ \ / | 18
\ X |
+------\---------------------\-+
F (to F) E D
at C the whole angle from CA down to CD is 90 degreesanswer 37
method At corner C the angle between the top line (towards A) and the side CD is a right angle, 90°. Three angles sit inside it, side by side: ∠ACF = 35°, then ∠FCE, then ∠ECD = 18°. So 35 + ∠FCE + 18 = 90, giving ∠FCE = 90 − 53 = 37°.
watch Adding the two given angles and stopping at 53, or subtracting from 180 instead of 90. The corner of a rectangle is 90°.
The 8-point compass shows the houses of 8 children around point O: Perry north, Ziming north-east, Bill east, Hadi south-east, Ali south, Sam south-west, Dave west, Shawn north-west. Roy is at point O facing Perry’s house. How many degrees must he turn in a clockwise direction so that he can face Shawn’s house?
Perry
Shawn | Ziming
\ | /
Dave -------O------- Bill
/ | \
Sam | Hadi
Alianswer 315
method Each step round an 8-point compass is 360 ÷ 8 = 45°. Starting at Perry (north) and going CLOCKWISE, count the steps to Shawn (north-west): Ziming, Bill, Hadi, Ali, Sam, Dave, Shawn — that is 7 steps. So the turn is 7 × 45 = 315°.
watch Turning the short way and answering 45°. That is anti-clockwise; the question asks for clockwise, which is the long way round.
Using a protractor, measure ∠ABC. BA runs horizontally to the left from B; BC rises steeply up and slightly to the right.
C
/
/
/
/
A ---------------------/ B
(angle here)
measure from BA, round to BCanswer 107
method Put the protractor's centre on B with its baseline along BA, then read where BC crosses the scale. BC leans back past the upright, so the angle is obtuse — a little over 90°, not under it. The reading is 107°.
watch Reading the wrong scale on the protractor and getting 73° instead. The two scales run in opposite directions, so decide FIRST whether the angle is acute or obtuse and pick the reading on that side of 90°. (A reading a degree or two either way is a fair measurement; the published key gives 107.)
Rectangle PQRS is folded to form the figure below. ∠y is three times ∠x. Find ∠x.
P Q +-------------------------------------+ | y / : | / : x | / : | \ / : +-----------------------------\-/-----+ S R the dashed line is where the corner R USED to be; the fold crease runs from Q
answer 18
method The whole corner at Q is 90°, and it is cut into three parts: ∠y, then the two angles the fold makes. A fold reflects, so the crease from Q sits exactly halfway between the edge in its new position and the edge in its old one — those two angles are equal, and each is ∠x. So y + x + x = 90. With y = 3x: 3x + 2x = 90, so 5x = 90 and x = 18°.
watch Writing y + x = 90 and getting 22.5°. The fold makes TWO equal angles below y, not one — that is the whole point of a crease.
A rectangular piece of paper PQRS was folded as shown below, so that the edge RQ came down onto the line RU. Find ∠TRU.
P T Q +---+---------------------------------+ | \ \ | \ \ U + \ \ | \ \ | \ \ 52 deg + - - - - - \- - - - - - + S R the dashed line S—R is where the bottom edge still is; TR is the crease
answer 19
method The corner of the rectangle at R is 90°, and it is now split into three: ∠QRT, ∠TRU and ∠URS. The fold carried RQ onto RU, and a crease always sits exactly halfway between where an edge was and where it went — so ∠QRT = ∠TRU. That leaves 90 = ∠TRU + ∠TRU + 52, so 2 × ∠TRU = 38 and ∠TRU = 19°.
watch Halving 52 instead. The 52° is the LEFTOVER angle outside the fold; the two equal angles are what remains of the 90° after taking it off.
Ben was standing at Point X. He turned 225° anti-clockwise and then made another 1/4-turn in the clockwise direction. He ended up facing the Bus Bay. Which location was Ben facing at first?
Library N
| ^
General Office | Bus Bay |
\ | /
Computer Lab ------- X ------- Field
/ | \
Dental Clinic | Basketball Court
|
Hall(1) Hall (2) Field (3) Library (4) Computer Lab
answer (1) — Hall
method Work out the NET turn first: 225° anti-clockwise, then 90° clockwise, leaves 225 − 90 = 135° anti-clockwise. He finished facing the Bus Bay (north-east), so to find where he began, undo it — turn 135° CLOCKWISE from north-east. North-east → (90°) → south-east → (45° more) → south. South is the Hall.
watch Applying the turns forwards from the Bus Bay instead of backwards. The Bus Bay is where he ENDED, so the working has to run in reverse.
A protractor is placed with its centre at Y and its baseline along the bottom. Two lines are drawn from Y: one up to the left towards X, reading 18° above the baseline, and one up to the right towards Z, reading 38° above the baseline. Find the value of ∠XYZ.
Z
X /
\ /
\ /
-----\--/--------- (the protractor baseline)
\/
Y
X is 18 deg above the baseline on the LEFT
Z is 38 deg above the baseline on the RIGHT(A) 18° (B) 38° (C) 124° (D) 142°
answer (C) — 124°
method The whole straight line along the protractor's base is 180°. The angle sits between the two drawn lines, so take off what lies outside it on each side: the 18° between YX and the baseline on the left, and the 38° between YZ and the baseline on the right. ∠XYZ = 180 − 18 − 38 = 124°.
watch Answering 142° by treating YX as lying ALONG the baseline (180 − 38). It does not — it is drawn 18° above it, and that 18° has to come off too.
Kayden is standing at point K facing the park. He makes a 135° clockwise turn followed by a 315° anticlockwise turn. Where will he be facing now?
Shopping Centre North
| ^
Library | Park |
\ | /
Cinema ---------K--------- Market
/ | \
Food Centre | Playground
|
School(A) Food Centre (B) Cinema (C) Market (D) School
answer (A) — Food Centre
method Combine the two turns first instead of doing them one at a time: 135° clockwise then 315° anticlockwise leaves 315 − 135 = 180° anticlockwise — which is the same as a half turn, so he ends up facing the exact opposite direction. He started facing the Park (north-east), and the opposite of north-east is south-west, which is the Food Centre.
watch Working the two turns separately and losing track. Also note a 180° turn needs no direction — clockwise or anticlockwise both land in the same place.
In the figure, ABCD is a rectangle and DEFG is a square, overlapping at D. ∠CDG = 64° and ∠EDH = 33°, where DH is a straight line from D. Find ∠ADH.
C D E
\ 64 / \ 33 /
\ / \ /
\ / G \ / H
\ / \
A
at D, the rays run in the order C, G, A, H, E
rectangle corner ADC = 90 deg · square corner GDE = 90 deganswer 31
method Use the two right angles the shapes give you. ABCD is a rectangle, so its corner at D is 90°: ∠CDA = 90. Since ∠CDG = 64, what is left is ∠GDA = 90 − 64 = 26°. DEFG is a square, so its corner at D is also 90°: ∠GDE = 90. Going from G round to E the rays pass A then H, so ∠GDA + ∠ADH + ∠HDE = 90, that is 26 + ∠ADH + 33 = 90. So ∠ADH = 90 − 59 = 31°.
watch Adding 64 and 33 and subtracting from 90 straight away. The 64° is measured inside the RECTANGLE's corner and the 33° inside the SQUARE's — two different right angles, so each has to be used on its own shape first.
The clock shown below was 45 minutes behind the actual time. To set the clock to the correct time, how many 1/4-turn(s) must the minute hand be moved clockwise?
12
11 1
10 2
|
9 +----> 3 minute hand on 12
hour hand on 3
8 4
7 5
6(1) 1 (2) 2 (3) 3 (4) 4
answer (3) — 3
method A full turn of the minute hand is 60 minutes, so a 1/4-turn is 60 ÷ 4 = 15 minutes. The clock is 45 minutes behind, so the minute hand must go forward 45 minutes. 45 ÷ 15 = 3 quarter-turns.
watch Moving the hand 45 minutes and answering 45, or working with the hour hand instead of the minute hand.
Helen was at a restaurant in a zoo. At the restaurant, Helen was facing south-west at first. She then made a 3/4-turn clockwise. Which animal would she be facing then?
Lion N
| ^
Elephant | Tiger |
\ | /
\ | /
Zebra ---- RESTAURANT ---- Leopard
/ | \
/ | \
Giraffe | Monkey
|
Penguin(1) Lion (2) Tiger (3) Monkey (4) Elephant
answer (3) — Monkey
method Each 1/4-turn is 90°. Going CLOCKWISE from south-west: 1/4-turn → north-west, 2/4-turn → north-east, 3/4-turn → south-east. On the map the south-east path leads to the Monkey.
watch Turning anticlockwise, which lands on north-west (the Elephant). Check which way the clock hands go before you start counting.