Question

1. a right triangle has a line of symmetry. select all conclusions that must be true.

a. all sides of the triangle have the same length.
b. all angles of the triangle have the same measure.
c. two sides of the triangle have the same length.
d. two angles of the triangle have the same measure.
e. no sides of the triangle have the same length.
f. no angles of the triangle have the same measure.
(from unit 1, lesson 15.)

1. in quadrilateral badc, ( ab = ad ) and ( bc = dc ). the

line ( ac ) is a line of symmetry for this
quadrilateral. based on the line of symmetry, explain
why angles ( acb ) and ( acd ) have the same measure.
(from unit 1, lesson 15.)

1. which of these constructions would construct a line of reflection that takes the point

a to point b?
a. construct the midpoint of segment ( ab ).
b. construct the perpendicular bisector of segment ( ab ).
c. construct a line tangent to circle a with radius ( ab ).
d. construct a vertical line passing through point a and a horizontal line passing
through point b.

Explanation:

Question 6

A right - triangle with a line of symmetry is an isosceles right - triangle. In an isosceles right - triangle, two sides (the legs) are of equal length. Also, the two non - right angles (each \(45^{\circ}\)) are of equal measure.

• Option A: In a right - triangle, by Pythagoras theorem \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the hypotenuse). If \(a = b\) (isosceles right - triangle), \(c=\sqrt{2}a

eq a\). So, not all sides are equal.

• Option B: A right - triangle has one angle equal to \(90^{\circ}\). If it is isosceles, the other two angles are \(45^{\circ}\) each. So, not all angles are equal.
• Option C: An isosceles right - triangle has two equal sides (the legs).
• Option D: An isosceles right - triangle has two equal non - right angles (each \(45^{\circ}\)).
• Option E: An isosceles right - triangle has two equal sides.
• Option F: An isosceles right - triangle has two equal non - right angles.

Since line \(AC\) is a line of symmetry for quadrilateral \(BADC\), a reflection over line \(AC\) maps quadrilateral \(BADC\) onto itself.
When we reflect a figure over a line of symmetry, corresponding parts of the figure are congruent. Point \(B\) is mapped to point \(D\) and point \(C\) is mapped to itself (because it lies on the line of symmetry \(AC\)).
The angle \(\angle ACB\) is mapped to \(\angle ACD\) under the reflection over line \(AC\). By the property of reflection (a type of congruence transformation), \(\angle ACB\cong\angle ACD\), so \(m\angle ACB = m\angle ACD\).

• Option A: The mid - point of segment \(AB\) is just a single point and does not define a line of reflection.
• Option B: The perpendicular bisector of segment \(AB\) is the set of points equidistant from \(A\) and \(B\). A reflection over the perpendicular bisector of segment \(AB\) will map point \(A\) to point \(B\). If \(l\) is the perpendicular bisector of \(AB\), for any point \(P\) on \(l\), \(PA = PB\) and the line \(l\) is perpendicular to \(AB\).
• Option C: A tangent to circle \(A\) with radius \(AB\) does not necessarily map \(A\) to \(B\).
• Option D: A vertical line through \(A\) and a horizontal line through \(B\) (unless \(A\) and \(B\) have a special position) does not map \(A\) to \(B\) under reflection.

Answer:

C. Two sides of the triangle have the same length; D. Two angles of the triangle have the same measure.

Question 7