Question

1. \\(\angle abd\\) is a right angle and \\(m\angle cbe = 35^\circ\\) as shown below. what is \\(m\angle ebd\\)?

a. \\(45^\circ\\)
b. \\(55^\circ\\)
c. \\(65^\circ\\)
d. \\(75^\circ\\)

1. when identifying the name of \\(\angle igf\\), alex wrote \\(\angle g\\) is obtuse. what is wrong with alexs statement and write the correct statement?

Explanation:

Identify given geometric relationships

We analyze the geometric configuration in Question 8.

• Points \(A\), \(B\), and \(C\) lie on a straight line, forming a straight angle \(\angle ABC = 180^\circ\).
• \(\angle ABD\) is a right angle, meaning \(m\angle ABD = 90^\circ\).
• Since \(\angle ABD\) and \(\angle CBD\) form a linear pair:

$$ m\angle CBD = 180^\circ - m\angle ABD = 90^\circ $$

Set up the angle equation

We use the Angle Addition Postulate for the adjacent angles that make up \(\angle CBD\).

• Ray \(BE\) lies in the interior of \(\angle CBD\).
• Therefore, we have:

$$ m\angle CBE + m\angle EBD = m\angle CBD $$

Solve for the unknown angle

We substitute the known values into our equation.

• Given: \(m\angle CBE = 35^\circ\) and \(m\angle CBD = 90^\circ\).

$$ 35^\circ + m\angle EBD = 90^\circ $$

$$ m\angle EBD = 90^\circ - 35^\circ = 55^\circ $$

Analyze the naming error in Question 10

We evaluate Alex's statement regarding \(\angle IGF\).

• Alex referred to \(\angle IGF\) simply as \(\angle G\).
• Vertex \(G\) is shared by multiple angles: \(\angle IGF\), \(\angle IGH\), and the straight angle \(\angle FGH\).
• Using a single letter to name an angle is only correct when there is no ambiguity (i.e., only one angle has that vertex).
• Since multiple angles share vertex \(G\), three letters must be used to specify the angle.
• Additionally, \(\angle IGF\) is acute, not obtuse.

Answer:

Question 8

• (A) \(45^\circ\)
• (B) \(55^\circ\) (Correct answer)
• (C) \(65^\circ\)
• (D) \(75^\circ\)

Question 10

Alex's statement is incorrect because vertex \(G\) is shared by multiple angles (\(\angle IGF\), \(\angle IGH\), and \(\angle FGH\)), so naming it simply \(\angle G\) is ambiguous. Additionally, \(\angle IGF\) is an acute angle, not an obtuse angle. The correct statement is that \(\angle IGF\) is acute.