Congruent and similar polygons questions with answers

Questions

1. Question 1Congruent quadrilaterals

   Quadrilateral $ABCD$ is congruent to quadrilateral $SPQR$. Find $x$.

   Show the worked solution

   1. Step 1: Match the vertices in the order of the statement$ABCD \equiv SPQR$: $S \leftrightarrow A$ and $P \leftrightarrow B$, so $SP$ matches $AB$.
   2. Step 2: Matching sides of congruent figures are equal$x = AB$$\phantom{x} = 7$
2. Question 2Similar polygons

   Quadrilateral $ABCD$ is similar to quadrilateral $PQRS$. Find $x$.

   Show the worked solution

   1. Step 1: Match the vertices in the order of the statement$ABCD \sim PQRS$: $A \leftrightarrow P$ and $D \leftrightarrow S$, so $\angle D = 96^\circ$. (Matching angles of similar figures are equal.)
   2. Step 2: The angles of the quadrilateral add up to 360°$x = 360 - 104 - 88 - 96$$x = 72$
3. Question 3Similar triangles sharing an angle

   In triangle $ABC$, $D$ lies on $AB$ and $E$ lies on $AC$ such that $\angle ADE = \angle ACB$. Find $x$.

   Show the worked solution

   1. Step 1: Name the similar triangles in matching order (∠A is shared)$\triangle ADE \sim \triangle ACB$ ($\angle A$ is common and $\angle ADE = \angle ACB$; AA)
   2. Step 2: Find AC from the ratio of matching sides$\dfrac{AC}{AD} = \dfrac{AB}{AE}$$AC = \dfrac{14}{10.5} \times 9$$\phantom{AC} = 12$
   3. Step 3: x = AC − AE$x = 12 - 10.5$$x = 1.5$
4. Question 4Similar triangles sharing a side

   In triangle $ABC$, $D$ lies on $AC$ such that $\angle ABD = \angle ACB$. $AC = 18$ cm. Find $x$.

   Show the worked solution

   1. Step 1: Name the similar triangles in matching order (∠A is shared)$\triangle ABD \sim \triangle ACB$ ($\angle A$ is common and $\angle ABD = \angle ACB$; AA)
   2. Step 2: Write the ratio of matching sides$\dfrac{AD}{AB} = \dfrac{AB}{AC}$$AD = \dfrac{12 \times 12}{18}$$AD = 8$
   3. Step 3: x = AC − AD$x = 18 - 8$$x = 10$
5. Question 5The scale factor of an enlargement

   On the squared paper, shape $B$ is an enlargement of shape $A$. Find the scale factor of the enlargement.

   Show the worked solution

   1. Step 1: Count the squares along a pair of matching sidesBottom side of shape $A$: $6$ units; of shape $B$: $12$ units
   2. Step 2: Scale factor = length on the image (B) ÷ length on the object (A)Scale factor $= \dfrac{12}{6}$Scale factor $= 2$
6. Question 6Congruent triangles and parallel lines

   In the diagram, the lines $AC$ and $BD$ cross at $M$, with $AM = MC$ and $BM = MD$. Which statement explains why $AB$ is parallel to $DC$?

   Show the worked solution

   1. Step 1: Show that triangles ABM and CDM are congruent$AM = CM$ (given), $\angle AMB = \angle CMD$ (vert. opp. ∠s), $BM = DM$ (given)$\triangle ABM \equiv \triangle CDM$ (SAS)
   2. Step 2: Matching angles of congruent triangles are equal$\angle BAM = \angle DCM$
   3. Step 3: What kind of angles are they?They are equal alternate angles, so $AB \parallel DC$.