Advanced Geometry: Angles, Coordinates, Transformations, and Shape Properties

Good to see you again. In the last lesson, you solved Questions 33–38 by making units match, choosing the right measurement formula, and checking whether an answer made sense in a real situation.

Now you will complete Questions 39–44 of the challenge. Geometry questions can look like drawing questions, but they are really reasoning questions: use a shape’s properties, record what stays the same, and work one step at a time. You will handle angles, coordinates, translations, reflections, and 3-D shape properties.

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A reliable geometry routine

Before calculating, identify the facts the diagram or wording gives you. These are the most useful facts for today:

Shape or situation	Fact to use
Full turn around a point	$360^{\circ }$
Straight line	$180^{\circ }$
Triangle	Angles total $180^{\circ }$
Rectangle	Every corner is $90^{\circ }$
Isosceles triangle	Two angles are equal
Translation	Shape slides; size, angles, and orientation stay the same
Reflection	Shape is flipped in a mirror line; each point stays the same distance from the line
Prism	Has two matching, parallel ends

A diagram may not be drawn accurately. Trust the labelled information and shape properties, not what “looks about right.”

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Angles: build the total from facts

Question 39: Isosceles triangle inside a rectangle

Rectangle $ABCD$ has $A$ at the bottom left and $B$ at the bottom right. An isosceles triangle $AEB$ is drawn inside it, with $AE=BE$. Angle $EAB$ is $68^{\circ }$.

Find:

• $x$, the angle $AEB$ at the top of the triangle;
• $y$, the angle between $BE$ and side $BC$ of the rectangle.

Because $AE=BE$, triangle $AEB$ is isosceles. Its two base angles are equal:

$$\angle EAB=\angle EBA=68^{\circ }$$

The three angles in a triangle total $180^{\circ }$, so:

$$x=180-68-68$$ $$x=44^{\circ }$$

Now look at corner $B$ of the rectangle. It is a right angle:

$$\angle ABC=90^{\circ }$$

That right angle has been split into $68^{\circ }$ and $y$:

$$68+y=90$$ $$y=22^{\circ }$$ $$\boxed{x=44^{\circ },y=22^{\circ }}$$

The important habit is to label the angle you have found immediately. It makes the next fact much easier to see.

Question 40: Equal angles around a point

Five angles meet at point $O$. Their sizes are:

$$40^{\circ },95^{\circ },x,x,x$$

Find $x$.

Angles around one point make one complete turn:

$$40+95+x+x+x=360$$

Collect the equal angles:

$$135+3x=360$$

Subtract $135$:

$$3x=225$$

Divide by $3$:

$$x=75^{\circ }$$ $$\boxed{75^{\circ }}$$

Check it:

$$40+95+75+75+75=360$$

The check works. Notice that writing $3x$ is just a quick way of writing three equal angles.

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Coordinates: position first, shape property second

A coordinate is written in the order $(x,y)$.

1. Read the $x$-coordinate first: move left or right.
2. Read the $y$-coordinate second: move down or up.

For example, $(-3,5)$ means $3$ units left and $5$ units up from $(0,0)$.

Question 41: Find a missing vertex of a parallelogram

The vertices of parallelogram $ABCD$ are given in order:

$$A=(-3,1),B=(2,1),C=(5,5)$$

Find the coordinates of $D$.

In a parallelogram, opposite sides are equal and parallel. Start by looking at the movement from $B$ to $C$:

• The $x$-coordinate changes from $2$ to $5$: move $3$ units right.
• The $y$-coordinate changes from $1$ to $5$: move $4$ units up.

To get from $A$ to $D$, make exactly the same movement:

$$(-3+3,1+4)$$ $$D=(0,5)$$ $$\boxed{D=(0,5)}$$

A useful check is that $D$ and $C$ should lie on the same horizontal line. They do: both have $y=5$.

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Transformations: change position without changing the shape

A translation slides a shape. Every vertex moves the same number of squares in the same direction.

A reflection flips a shape across a mirror line. Each original point and its reflected point must be:

• the same perpendicular distance from the mirror line;
• at the same height if the mirror line is vertical;
• at the same left-right position if the mirror line is horizontal.

Question 42: Translate, then reflect

Triangle $ABC$ has these vertices:

$$A=(-4,2),B=(-1,2),C=(-2,5)$$

First, translate the triangle $4$ units right and $3$ units down. Then reflect the new triangle in the $y$-axis.

Find the final coordinates of $A$, $B$, and $C$.

For the translation:

• $4$ units right means add $4$ to each $x$-coordinate.
• $3$ units down means subtract $3$ from each $y$-coordinate.

Vertex	Original coordinate	After translation
$A$	$(-4,2)$	$(0,-1)$
$B$	$(-1,2)$	$(3,-1)$
$C$	$(-2,5)$	$(2,2)$

Now reflect in the $y$-axis. A reflection in the $y$-axis changes the sign of the $x$-coordinate, while the $y$-coordinate stays unchanged.

Vertex	Before reflection	Final coordinate
$A$	$(0,-1)$	$(0,-1)$
$B$	$(3,-1)$	$(-3,-1)$
$C$	$(2,2)$	$(-2,2)$

$$\boxed{A=(0,-1),B=(-3,-1),C=(-2,2)}$$

Point $A$ did not move during the reflection because it lies on the $y$-axis. Points on a mirror line always remain where they are.

Question 43: Reflect in a line that is not an axis

Triangle $PQR$ has vertices:

$$P=(5,1),Q=(7,1),R=(6,4)$$

Reflect it in the vertical mirror line $x=2$. Find the new coordinates.

Do not simply change the signs of the $x$-coordinates: that method works only when the mirror line is the $y$-axis. Instead, count each point’s distance from $x=2$.

For $P=(5,1)$:

• $P$ is $3$ units right of the mirror line because $5-2=3$.
• Its reflection must be $3$ units left of $x=2$.
• $2-3=-1$.

So the reflection of $P$ is:

$$P^{\prime }=(-1,1)$$

For $Q=(7,1)$:

• $Q$ is $5$ units right of the mirror line.
• Its reflection is $5$ units left of the mirror line.

$$Q^{\prime }=(-3,1)$$

For $R=(6,4)$:

• $R$ is $4$ units right of the mirror line.
• Its reflection is $4$ units left of the mirror line.

$$R^{\prime }=(-2,4)$$ $$\boxed{P^{\prime }=(-1,1),Q^{\prime }=(-3,1),R^{\prime }=(-2,4)}$$

The $y$-coordinates stayed the same because the mirror line was vertical. The reflected triangle has exactly the same side lengths and angles as the original one.

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Properties of 3-D shapes

A 3-D shape has:

• faces: its flat surfaces;
• edges: the line segments where faces meet;
• vertices: its corners.

A prism has two matching bases. A pyramid has one base and an apex, so do not count its faces, edges, and vertices in the same way.

Question 44: A hexagonal prism

A solid has two matching hexagonal faces. These are joined by six rectangular faces. It is a hexagonal prism.

How many faces, vertices, and edges does it have?

Start with the faces:

• $2$ hexagonal faces;
• $6$ rectangular faces.

$$2+6=8$$

So it has:

$$\boxed{8\text{ faces}}$$

Each hexagonal end has $6$ vertices. There are two ends:

$$6+6=12$$

So it has:

$$\boxed{12\text{ vertices}}$$

Now count edges carefully:

• $6$ edges around the first hexagon;
• $6$ edges around the second hexagon;
• $6$ joining edges between the two hexagons.

$$6+6+6=18$$ $$\boxed{18\text{ edges}}$$

A simple way to check is to imagine making the frame from sticks: you would need $18$ sticks.

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Key takeaways

Geometry becomes manageable when you turn each diagram into a set of facts.

• Use $180^{\circ }$ for triangles and straight lines, $360^{\circ }$ around a point, and $90^{\circ }$ at each rectangle corner.
• In an isosceles triangle, the two equal sides face the two equal angles.
• Coordinates are always written $x$ first, then $y$.
• In a parallelogram, opposite sides have the same movement across the grid.
• For a translation, move every vertex by the same amount.
• For a reflection, every vertex must end the same distance from the mirror line.
• For 3-D shapes, count faces, vertices, and edges systematically rather than guessing from a drawing.

Next, you will finish the 50-question challenge with Questions 45–50, combining data handling with mixed multi-step reasoning.