Transformations and congruence

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1. Geometric constructions with compass and straightedge

Warm-up

Three quick questions from earlier lessons.

Discover

With only a compass and a straightedge (no ruler marks), can you find the exact midpoint of a segment?

  1. Yes, with two equal arcs
  2. No, you need to measure

Key idea

Constructions use equal distances, not measurements.

  1. Step 1The angle at O.
  2. Step 21. An arc centered at O crosses the sides at A and B.
  3. Step 32. Equal arcs from A and from B meet at P.
  4. Step 43. The ray OP bisects the angle (SSS: OAP ≅ OBP).

A compass keeps a distance fixed; a straightedge draws lines through points you already have. With them you can copy a segment or an angle, bisect a segment or an angle, draw perpendicular and parallel lines, and inscribe a regular hexagon (step the radius 6 times around the circle), an equilateral triangle (every other hexagon vertex) or a square (two perpendicular diameters). Each step is justified by congruent triangles or equal radii.

Worked example

Bisect the angle at vertex O.

it crosses the sides at A and B, with OA = OB
Arc from O.
they meet at P, with AP = BP
Equal arcs from A and B.
triangles OAP and OBP are congruent (SSS), so OP bisects the angle
Draw OP.

Your turn, guided

Your turn: a regular hexagon inside a circle of radius 5 cm. Each side?

55 cm
The side equals the radius.

Practice ★

  1. Put the steps for constructing the perpendicular bisector of a segment AB of length 7 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  2. Put the steps for constructing the perpendicular bisector of a segment AB of length 9 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  1. To inscribe a regular hexagon in a circle of radius 8 cm, you step the compass around the circle. Which opening do you keep?
    1. 8 cm, the radius of the circle
    2. 16 cm, the diameter
    3. Any opening, as long as it never changes

Watch out!

Practice ★★

  1. In the construction of the perpendicular bisector of AB (AB = 10 cm), why does the line PQ through the two arc crossings bisect AB at a right angle?
    1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
    2. Because the angle measures 90° with a protractor
    3. Because the arcs have a radius of 10 cm
  2. In the construction of the perpendicular bisector of AB (AB = 7 cm), why does the line PQ through the two arc crossings bisect AB at a right angle?
    1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
    2. Because the angle measures 90° with a protractor
    3. Because the arcs have a radius of 7 cm
  1. A regular hexagon is constructed inside a circle of radius 4 cm, with its vertices on the circle. What is its perimeter, in cm?Not to scale

Exit ticket

Three questions to check you have got it.

  1. Put the steps for constructing the perpendicular bisector of a segment AB of length 7 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  2. Put the steps for constructing the perpendicular bisector of a segment AB of length 9 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge
  3. Put the steps for constructing the perpendicular bisector of a segment AB of length 6 cm in order.
    • With the compass point on A, draw an arc with a radius of more than half of AB
    • Keep the same opening; with the point on B, draw an arc that crosses the first one twice
    • Label the two crossing points P and Q
    • Draw the line through P and Q with the straightedge

Summary card

Constructions

Compass for equal distances, straightedge for lines; justify with congruent triangles.

Watch out: do not change the compass opening between the two arcs.

2. Translations and rigid transformations

Warm-up

Three quick questions from earlier lessons.

Discover

A triangle slides 3 units right and 2 units down without turning. What happens to its side lengths?

  1. They stay the same
  2. They grow by 3

Key idea

A translation moves every point by the same directed distance.

A translation by (a,b)(a, b) sends (x,y)(x, y) to (x+a,y+b)(x + a, y + b). Translations, reflections and rotations are rigid transformations: they preserve lengths and angle measures.

Worked example

Translate A(2,−1)A(2, -1) by 4 left and 3 up.

  1. Step 1Start at A(2, −1).
  2. Step 2a = −4: move 4 to the left.
  3. Step 3b = 3: move 3 up.
  4. Step 4The image is A′(2 − 4, −1 + 3) = (−2, 2).

a=−4a = -4, b=3b = 3
Left is negative x.
A′=(2−4,−1+3)A' = (2 - 4, -1 + 3)
Add.
A′=(−2,2)A' = (-2, 2)
Compute.

Your turn, guided

Your turn: the y-coordinate of (5,1)(5, 1) translated 6 down.

−5-5
1 − 6.

Practice ★

  1. Translate the point A(−4,−4)A(-4, -4) by 2 units right and 1 units down. Tap its image A′A'.
  2. Translate the point A(1,0)A(1, 0) by 4 units right and 4 units down. Tap its image A′A'.
  3. Translate the point A(2,−1)A(2, -1) by 1 units right and 2 units down. Tap its image A′A'.

Watch out!

Practice ★★

  1. A translation takes (−3,0)(-3, 0) to (0,5)(0, 5). By how many units does it move points horizontally (positive = right)?
  2. A translation takes (−2,4)(-2, 4) to (−4,4)(-4, 4). By how many units does it move points horizontally (positive = right)?
  1. A polygon with 9 sides is translated. What changes?
    1. Only its position: lengths and angles stay the same
    2. Its side lengths
    3. Its angles
  1. Translate triangle ABC 2 right and 2 down. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

  1. Translate the point A(−4,−4)A(-4, -4) by 2 units right and 1 units down. Tap its image A′A'.
  2. Translate the point A(1,0)A(1, 0) by 4 units right and 4 units down. Tap its image A′A'.
  3. Translate the point A(2,−1)A(2, -1) by 1 units right and 2 units down. Tap its image A′A'.

Summary card

Translations

(x,y)→(x+a,y+b)(x, y) \to (x + a, y + b); rigid: lengths and angles kept.

Watch out: left and down are negative.

3. Reflections in the coordinate plane

Warm-up

Three quick questions from earlier lessons.

Discover

A point is 3 units above the xx-axis. Its mirror image across the xx-axis is…

  1. 3 units below the axis, straight down
  2. 3 units to the left

Key idea

The line of reflection is the perpendicular bisector of each segment joining a point to its image.

Across the xx-axis: (x,y)→(x,−y)(x, y) \to (x, -y). Across the yy-axis: (x,y)→(−x,y)(x, y) \to (-x, y). Across y=xy = x: (x,y)→(y,x)(x, y) \to (y, x).

Worked example

Reflect B(−2,5)B(-2, 5) in three lines.

  1. Step 1Triangle ABC above the x-axis.
  2. Step 2From each vertex, go straight across the axis, the same distance on the other side.
  3. Step 3Join the images: (x, y) → (x, −y).

xx-axis: (−2,−5)(-2, -5)
Negate y.
yy-axis: (2,5)(2, 5)
Negate x.
y=xy = x: (5,−2)(5, -2)
Swap.

Your turn, guided

Your turn: the x-coordinate of (4,−3)(4, -3) reflected across the yy-axis.

−4-4
Negate x.

Practice ★

  1. Reflect A(3,2)A(3, 2) across the xx-axis. Tap its image.
  2. Reflect A(5,6)A(5, 6) across the xx-axis. Tap its image.
  3. Reflect A(−4,1)A(-4, 1) across the xx-axis. Tap its image.

Watch out!

Practice ★★

  1. Reflect A(5,−1)A(5, -1) across the line y=xy = x. Tap its image.
  2. Reflect A(−5,−3)A(-5, -3) across the line y=xy = x. Tap its image.
  1. A point is 5 units from a line of reflection. How far is it from its image?
  1. Reflect triangle ABC across the yy-axis. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

  1. Reflect A(3,2)A(3, 2) across the xx-axis. Tap its image.
  2. Reflect A(5,6)A(5, 6) across the xx-axis. Tap its image.
  3. Reflect A(−4,1)A(-4, 1) across the xx-axis. Tap its image.

Summary card

Reflections

xx-axis: (x,−y)(x, -y); yy-axis: (−x,y)(-x, y); y=xy = x: (y,x)(y, x).

Watch out: across the xx-axis, it is yy that changes sign.

4. Rotations about the origin

Warm-up

Three quick questions from earlier lessons.

Discover

Rotate the point (1,0)(1, 0) by 90∘90^\circ counterclockwise about the origin. Where does it go?

  1. (0,1)(0, 1)
  2. (0,−1)(0, -1)

Move P and change the angle. Compare OP and OQ.

  1. They are always equal
  2. OQ grows with the angle

Key idea

Rotations about the origin by quarter turns.

About the origin: 90∘90^\circ counterclockwise: (x,y)→(−y,x)(x, y) \to (-y, x); 180∘180^\circ: (x,y)→(−x,−y)(x, y) \to (-x, -y); 90∘90^\circ clockwise (270∘270^\circ counterclockwise): (x,y)→(y,−x)(x, y) \to (y, -x).

Worked example

Rotate C(3,2)C(3, 2).

  1. Step 1Triangle ABC.
  2. Step 2Turn it about O: every point keeps its distance to O.
  3. Step 3A quarter turn (90° counterclockwise): (x, y) → (−y, x).

90∘90^\circ ccw: (−2,3)(-2, 3)
(−y, x).
180∘180^\circ: (−3,−2)(-3, -2)
(−x, −y).
90∘90^\circ cw: (2,−3)(2, -3)
(y, −x).

Your turn, guided

Your turn: the y-coordinate of (5,1)(5, 1) after 90∘90^\circ counterclockwise.

55
(−y, x) = (−1, 5).

Practice ★

  1. Rotate A(−2,−1)A(-2, -1) by 180∘180^\circ about the origin. Tap its image.
  2. Rotate A(3,−2)A(3, -2) by 180∘180^\circ about the origin. Tap its image.
  1. Rotate A(1,3)A(1, 3) by 90∘90^\circ counterclockwise about the origin. Tap its image.
  2. Rotate A(1,2)A(1, 2) by 90∘90^\circ counterclockwise about the origin. Tap its image.

Watch out!

Practice ★★

  1. A figure is rotated 7 times by 90∘90^\circ in the same direction. What single rotation, between 0∘0^\circ and 360∘360^\circ, gives the same result? (degrees)
  1. Boss problem. A=(4,6)A = (4, 6) is reflected across the xx-axis, then rotated 90∘90^\circ counterclockwise about the origin, then translated 5 units right.
  1. Rotate triangle ABC by 90° counterclockwise about the origin. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

  1. Rotate A(1,3)A(1, 3) by 90∘90^\circ counterclockwise about the origin. Tap its image.
  2. Rotate A(1,2)A(1, 2) by 90∘90^\circ counterclockwise about the origin. Tap its image.
  3. Rotate A(2,6)A(2, 6) by 90∘90^\circ counterclockwise about the origin. Tap its image.

Summary card

Rotations about the origin

ccw 90°: (−y,x)(-y, x); 180°: (−x,−y)(-x, -y); cw 90°: (y,−x)(y, -x).

Watch out: check the quadrant the image lands in.

5. Symmetry

Warm-up

Three quick questions from earlier lessons.

Discover

A square is turned about its center. Which is the smallest turn that makes it look exactly the same?

  1. 90∘90^\circ
  2. 180∘180^\circ

Key idea

Symmetries are rigid transformations that map a figure onto itself.

A figure has reflection symmetry across a line if reflecting maps it onto itself, and rotational symmetry if a rotation of less than 360∘360^\circ does. A regular polygon with nn sides has nn lines of symmetry and rotational symmetry by multiples of 360∘n\tfrac{360^\circ}{n}.

Worked example

A regular hexagon.

6 lines of symmetry
One through each vertex (pairs with the opposite one) and each edge midpoint.
Smallest rotation: 60∘60^\circ
360° ÷ 6.

Your turn, guided

Your turn: the smallest rotation for a regular pentagon.

72∘72^\circ
360 ÷ 5.

Practice ★

  1. What is the smallest positive angle of rotation that maps a regular polygon with 8 sides onto itself? (degrees)Not to scale
  2. What is the smallest positive angle of rotation that maps a regular polygon with 3 sides onto itself? (degrees)Not to scale
  1. How many lines of symmetry does a regular polygon with 4 sides have?Not to scale

Watch out!

Practice ★★

  1. A rectangle 4 cm by 6 cm (not a square). Which transformations map it onto itself?
    1. Reflections across its two midlines, and a 180° rotation about its center
    2. Reflections across its diagonals
    3. A 90° rotation

Exit ticket

Three questions to check you have got it.

  1. What is the smallest positive angle of rotation that maps a regular polygon with 8 sides onto itself? (degrees)Not to scale
  2. What is the smallest positive angle of rotation that maps a regular polygon with 3 sides onto itself? (degrees)Not to scale
  3. What is the smallest positive angle of rotation that maps a regular polygon with 11 sides onto itself? (degrees)Not to scale

Summary card

Symmetry

Regular nn-gon: nn lines; rotations by 360∘/n360^\circ/n.

Watch out: diagonals of a rectangle are not lines of symmetry.

6. Angle relationships: vertical angles, parallel lines, triangle sum

Warm-up

Three quick questions from earlier lessons.

Discover

Two lines cross, making four angles. One is 70∘70^\circ. What is the angle opposite it?

  1. 70∘70^\circ
  2. 110∘110^\circ

Drag the corners of the triangle. Watch the three angles.

  1. They always add up to 180°
  2. It depends on the shape of the triangle

Key idea

Angle facts, proved with rigid transformations.

Vertical angles are equal. Angles on a straight line add to 180∘180^\circ. With a transversal across parallel lines: corresponding and alternate interior angles are equal; same-side interior angles add to 180∘180^\circ. The angles of a triangle add to 180∘180^\circ.

Worked example

A triangle has angles xx, 2x2x and 60∘60^\circ.

x+2x+60=180x + 2x + 60 = 180
Angle sum.
3x=1203x = 120, x=40x = 40
Solve.
40∘40^\circ, 80∘80^\circ, 60∘60^\circ
The angles.

Your turn, guided

Your turn: a straight angle is split into 115∘115^\circ and yy.

y=65∘y = 65^\circ
180 − 115.

Practice ★

  1. Two angles of a triangle measure 32∘32^\circ and 20∘20^\circ. Find the third angle, in degrees.
  2. Two angles of a triangle measure 50∘50^\circ and 35∘35^\circ. Find the third angle, in degrees.
  1. Two lines cross. One angle measures 30∘30^\circ. Find the angle vertically opposite it, in degrees.

Watch out!

Practice ★★

  1. A transversal crosses two parallel lines and makes a 33∘33^\circ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.
  2. A transversal crosses two parallel lines and makes a 80∘80^\circ angle with the first. Find the co-interior (same-side interior) angle it makes with the second line, in degrees.

Exit ticket

Three questions to check you have got it.

  1. Two angles of a triangle measure 32∘32^\circ and 20∘20^\circ. Find the third angle, in degrees.
  2. Two angles of a triangle measure 50∘50^\circ and 35∘35^\circ. Find the third angle, in degrees.
  3. Two angles of a triangle measure 41∘41^\circ and 80∘80^\circ. Find the third angle, in degrees.

Summary card

Angle relationships

Vertical: equal. Straight line: 180°. Parallel lines: corresponding/alternate equal, same-side interior 180°. Triangle: 180°.

Watch out: equal or supplementary? Picture the rotation.

7. Congruence and corresponding parts

Warm-up

Three quick questions from earlier lessons.

Discover

△ABC≅△DEF\triangle ABC \cong \triangle DEF. Which angle of △DEF\triangle DEF matches ∠B\angle B?

  1. ∠E\angle E
  2. ∠F\angle F

Key idea

Congruent figures match part for part.

Two figures are congruent if a sequence of rigid transformations maps one onto the other. Then corresponding sides and angles are equal. In △ABC≅△DEF\triangle ABC \cong \triangle DEF the order of letters gives the correspondence: A↔DA \leftrightarrow D, B↔EB \leftrightarrow E, C↔FC \leftrightarrow F.

Worked example

△PQR≅△STU\triangle PQR \cong \triangle STU, PQ=7PQ = 7, ∠R=40∘\angle R = 40^\circ.

ST=7ST = 7
P↔S, Q↔T.
∠U=40∘\angle U = 40^\circ
R↔U.

Your turn, guided

Your turn: △ABC≅△XYZ\triangle ABC \cong \triangle XYZ and BC=12BC = 12. Find YZYZ.

YZ=12YZ = 12
B↔Y, C↔Z.

Practice ★

  1. △ABC≅△DEF\triangle ABC \cong \triangle DEF with AB=3AB = 3, BC=11BC = 11 and CA=18CA = 18. Find DEDE.
  2. △ABC≅△DEF\triangle ABC \cong \triangle DEF with AB=9AB = 9, BC=15BC = 15 and CA=16CA = 16. Find DEDE.
  1. △KLM≅△STU\triangle KLM \cong \triangle STU. Which segment corresponds to LMLM (length 5)?
    1. TUTU
    2. STST
    3. SUSU

Watch out!

Practice ★★

  1. △PQR≅△XYZ\triangle PQR \cong \triangle XYZ, with ∠Q=43∘\angle Q = 43^\circ and ∠R=40∘\angle R = 40^\circ. Find ∠X\angle X, in degrees.
  2. △PQR≅△XYZ\triangle PQR \cong \triangle XYZ, with ∠Q=37∘\angle Q = 37^\circ and ∠R=47∘\angle R = 47^\circ. Find ∠X\angle X, in degrees.

Exit ticket

Three questions to check you have got it.

  1. △ABC≅△DEF\triangle ABC \cong \triangle DEF with AB=3AB = 3, BC=11BC = 11 and CA=18CA = 18. Find DEDE.
  2. △ABC≅△DEF\triangle ABC \cong \triangle DEF with AB=9AB = 9, BC=15BC = 15 and CA=16CA = 16. Find DEDE.
  3. △ABC≅△DEF\triangle ABC \cong \triangle DEF with AB=4AB = 4, BC=11BC = 11 and CA=17CA = 17. Find DEDE.

Summary card

Corresponding parts

The order of letters in a congruence statement gives the matching.

Watch out: match by position, not by appearance.

8. Triangle congruence criteria

Warm-up

Three quick questions from earlier lessons.

Discover

You know two sides of a triangle, 5 cm and 7 cm, and the 40∘40^\circ angle between them. How many different triangles can you draw?

  1. Just one (up to position)
  2. Infinitely many

Key idea

Three matching parts can be enough, if they are the right ones.

Triangles are congruent if they match by SSS, SAS (the angle between the sides), ASA or AAS. SSA (angle not between) and AAA do not guarantee congruence.

Worked example

Which criterion?

3 pairs of equal sides: SSS
All sides.
2 angles and the side between them: ASA
Side between two angles.
2 sides and a non-included angle: no conclusion
Not a criterion.

Your turn, guided

Your turn: how many of SSS, SAS, ASA, AAS, SSA, AAA are valid criteria?

44
SSA and AAA are not.

Practice ★

  1. Two triangles have two pairs of equal sides (3 cm and 5 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  1. Two triangles both have angles 48∘48^\circ, 60∘60^\circ and 72∘72^\circ. Are they congruent?
    1. Not necessarily: they have the same shape but may differ in size
    2. Yes, all three angles match
    3. No, never

Watch out!

Practice ★★

  1. Two triangles have two equal pairs of sides (3 and 6) and an equal angle not between them. Must they be congruent?
    1. Not necessarily: side-side-angle can give two different triangles
    2. Yes, by SSA
    3. Yes, by SAS

Exit ticket

Three questions to check you have got it.

  1. Two triangles have two pairs of equal sides (3 cm and 5 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  2. Two triangles have two pairs of equal sides (9 cm and 11 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA
  3. Two triangles have two pairs of equal sides (5 cm and 7 cm) and the angles between those sides equal. Which criterion shows they are congruent?
    1. SAS
    2. SSA
    3. AAA

Summary card

Congruence criteria

SSS, SAS, ASA, AAS. Not SSA, not AAA.

Watch out: in SAS the angle is between the sides.

Answers: Transformations and congruence

1. Geometric constructions with compass and straightedge

A

  1. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  2. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge

B

  1. 8 cm, the radius of the circle

C

  1. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector
  2. P and Q are each as far from A as from B, and such points lie on the perpendicular bisector

D

  1. 2424

E

  1. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  2. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge
  3. 1. With the compass point on A, draw an arc with a radius of more than half of AB 2. Keep the same opening; with the point on B, draw an arc that crosses the first one twice 3. Label the two crossing points P and Q 4. Draw the line through P and Q with the straightedge

2. Translations and rigid transformations

A

  1. (−2,−5)(-2, -5)
  2. (5,−4)(5, -4)
  3. (3,−3)(3, -3)

B

  1. 33
  2. −2-2

C

  1. Only its position: lengths and angles stay the same

D

  1. (−3,−2), (0,−2), (−2,0)(-3, -2),\ (0, -2),\ (-2, 0)

E

  1. (−2,−5)(-2, -5)
  2. (5,−4)(5, -4)
  3. (3,−3)(3, -3)

3. Reflections in the coordinate plane

A

  1. (3,−2)(3, -2)
  2. (5,−6)(5, -6)
  3. (−4,−1)(-4, -1)

B

  1. (−1,5)(-1, 5)
  2. (−3,−5)(-3, -5)

C

  1. 1010

D

  1. (−4,3), (−7,3), (−5,5)(-4, 3),\ (-7, 3),\ (-5, 5)

E

  1. (3,−2)(3, -2)
  2. (5,−6)(5, -6)
  3. (−4,−1)(-4, -1)

4. Rotations about the origin

A

  1. (2,1)(2, 1)
  2. (−3,2)(-3, 2)

B

  1. (−3,1)(-3, 1)
  2. (−2,1)(-2, 1)

C

  1. 270270

D

  1. 1111

E

  1. (−3,3), (−3,6), (−5,4)(-3, 3),\ (-3, 6),\ (-5, 4)

F

  1. (−3,1)(-3, 1)
  2. (−2,1)(-2, 1)
  3. (−6,2)(-6, 2)

5. Symmetry

A

  1. 4545
  2. 120120

B

  1. 44

C

  1. Reflections across its two midlines, and a 180° rotation about its center

D

  1. 4545
  2. 120120
  3. 36011\frac{360}{11}

6. Angle relationships: vertical angles, parallel lines, triangle sum

A

  1. 128128
  2. 9595

B

  1. 3030

C

  1. 147147
  2. 100100

D

  1. 128128
  2. 9595
  3. 5959

7. Congruence and corresponding parts

A

  1. 33
  2. 99

B

  1. TUTU

C

  1. 9797
  2. 9696

D

  1. 33
  2. 99
  3. 44

8. Triangle congruence criteria

A

  1. SAS

B

  1. Not necessarily: they have the same shape but may differ in size

C

  1. Not necessarily: side-side-angle can give two different triangles

D

  1. SAS
  2. SAS
  3. SAS