the angle bisectors of $\triangle abc$ are $\overline{av}$, $\overline{bv}$, and $\overline{cv}$. they meet at a single point $v$.
(in other words, $v$ is the incenter of $\triangle abc$.)

suppose $tv = 18$, $cv = 23$, $m\angle tcu = 44^\circ$, and $m\angle sav = 52^\circ$.
find the following measures.
note that the figure is not drawn to scale.

$m\angle sau = \square^\circ$
$m\angle sbv = \square^\circ$
$sv = \square$

Question

the angle bisectors of $\triangle abc$ are $\overline{av}$, $\overline{bv}$, and $\overline{cv}$. they meet at a single point $v$.
(in other words, $v$ is the incenter of $\triangle abc$.)

suppose $tv = 18$, $cv = 23$, $m\angle tcu = 44^\circ$, and $m\angle sav = 52^\circ$.
find the following measures.
note that the figure is not drawn to scale.

$m\angle sau = \square^\circ$
$m\angle sbv = \square^\circ$
$sv = \square$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Incenter Properties" ], "current_concepts": [ "Incenter Theorem", "Angle Bisector Theorem", "Incenter Properties" ] } ## Find the measure of angle SAU $$ \begin{aligned} &m\angle SAV = 52^\circ\ &\text{Since } AV \text{ is the angle bisector of } \angle A:\ &m\angle UAV = m\angle SAV = 52^\circ\ &m\angle SAU = m\angle SAV + m\angle UAV = 52^\circ + 52^\circ = 104^\circ \end{aligned} $$ ## Find the measure of angle SBV $$ \begin{aligned} &m\angle SAU = 104^\circ\ &m\angle TCU = 44^\circ \implies m\angle C = 44^\circ\ &m\angle B = 180^\circ - m\angle SAU - m\angle C = 180^\circ - 104^\circ - 44^\circ = 32^\circ\ &\text{Since } BV \text{ is the angle bisector of } \angle B:\ &m\angle SBV = \frac{m\angle B}{2} = \frac{32^\circ}{2} = 16^\circ \end{aligned} $$ ## Find the length of SV $$ \begin{aligned} &\text{By the Incenter Theorem, the incenter } V \text{ is equidistant from all three sides:}\ &SV = TV = UV\ &\text{Given } TV = 18:\ &SV = 18 \end{aligned} $$ - $m\angle SAU =$ 104$^\circ$ - $m\angle SBV =$ 16$^\circ$ - $SV =$ 18 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Incenter Theorem" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the measure of angle SAU

Find the measure of angle SBV

Find the length of SV

Answer: