Question

directions: $\angle abc$ is bisected by $\overrightarrow{bd}$. find the value of $x$.
(10) diagram: $a\leftarrow----b----\
ightarrow c$, $d$ above $b$; $\angle abd = (3x + 1)\degree$, $\angle dbc = 76\degree$
(11) diagram: $a\leftarrow----b$, $c$ above $b$, $d$ between $a$ and $c$; $\angle abd = (8x)\degree$, $\angle dbc = 40\degree$
(12) diagram: $a\leftarrow----b$, $c$ above $b$ (vertical), $d$ between $a$ and $c$; $\angle abd = (2x - 1)\degree$, $\angle dbc = 45\degree$

Explanation:

Problem (10)

Step1: Use angle bisector property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), \(\angle ABD=\angle DBC\). So \((3x + 1)^\circ=76^\circ\).

Step2: Solve for \(x\)

\[

$$\begin{align*} 3x+1&=76\\ 3x&=76 - 1\\ 3x&=75\\ x&=\frac{75}{3}\\ x& = 25 \end{align*}$$

\]

Step1: Use angle bisector property

\(\overrightarrow{BD}\) bisects \(\angle ABC\), so \(\angle ABD=\angle DBC\). Thus, \((8x)^\circ = 40^\circ\).

Step2: Solve for \(x\)

\[

$$\begin{align*} 8x&=40\\ x&=\frac{40}{8}\\ x&=5 \end{align*}$$

\]

Step1: Analyze right angle and bisector

\(\angle ABC\) is a right angle? Wait, \(\angle DBC = 45^\circ\), and \(\overrightarrow{BD}\) bisects \(\angle ABC\), so \(\angle ABD=\angle DBC\). Thus, \((2x - 1)^\circ=45^\circ\).

Step2: Solve for \(x\)

\[

$$\begin{align*} 2x-1&=45\\ 2x&=45 + 1\\ 2x&=46\\ x&=\frac{46}{2}\\ x&=23 \end{align*}$$

\]

Answer:

\(25\)

Problem (11)