Lesson 1

Lines, Angles, and Curves

  • Let’s define some line segments and angles related to circles.

Problem 1

Find the values of \(x, y,\) and \(z\).

Circle with center A.

Problem 2

Give an example from the image of each kind of segment.

  1. a diameter
  2. a chord that is not a diameter
  3. a radius
Circle with center A.

Problem 3

Identify whether each statement must be true, could possibly be true, or definitely can’t be true.

  1. A diameter is a chord.
  2. A radius is a chord.
  3. A chord is a diameter.
  4. A central angle measures 90\(^\circ\).

Problem 4

Write an equation of the altitude from vertex \(A\).

Triangle ABC graphed. A = 2 comma 1, B = -1 comma 4, C = 8 comma 2. altitude drawn from vertex A to side BC.


(From Unit 6, Lesson 17.)

Problem 5

Triangle \(ABC\) has vertices at \((5,0), (1,6),\) and \((9,3)\). What is the point of intersection of the triangle’s medians?


The medians do not intersect in a single point.







(From Unit 6, Lesson 16.)

Problem 6

Consider the parallelogram with vertices at \((0,0), (8,0), (4,6),\) and \((12,6)\). Where do the diagonals of this parallelogram intersect?

(From Unit 6, Lesson 15.)

Problem 7

Lines \(\ell\) and \(p\) are parallel. Select all true statements.

\(\text{Parallel lines }l \text{ and }p \text{ on a coordinate plane, origin } O\)

Triangle \(ADB\) is congruent to triangle \(CEF\).


The slope of line \(\ell\) is equal to the slope of line \(p\).


Triangle \(ADB\) is similar to triangle \(CEF\).


\(\sin(A) = \sin(C)\)


\(\cos(B) = \sin(C)\)

(From Unit 6, Lesson 10.)

Problem 8

Mai wrote a proof that triangle \(AED\) is congruent to triangle \(CEB\). Mai's proof is incomplete. How can Mai fix her proof?

We know side \(AE\) is congruent to side \(CE\) and angle \(A\) is congruent to angle \(C\). By the Angle-Side-Angle Triangle Congruence Theorem, triangle \(AED\) is congruent to triangle \(CEB\).

\(\angle A \cong \angle C, \overline{AE} \cong \overline{CE}\)

Triangles ADE and CBE meet only at point E. Angles A and C are congruent. Sides AE and CE are congruent.
(From Unit 2, Lesson 7.)