Exercise 9. Let $A B C$ be a triangle such that $\widehat{C A B}=20^{\circ}$. Let $D$ be the midpoint of the segment $[A B]$. Suppose that $\widehat{C D B}=40^{\circ}$. What is the value of the angle $\widehat{A B C}$?
## Solution to Exercise 9

![](https://cdn.mathpix.com/cropped/2024_06_04_0d4f01d8013de0f79932g-15.jpg?height=465&width=1250&top_left_y=567&top_left_x=423)

Since $\widehat{C D B}=40^{\circ}$, we obtain by supplementary angles that $\widehat{C D A}=180-\widehat{C D B}=140^{\circ}$. Since the sum of the angles in triangle $A D C$ is $180^{\circ}$, we get $\widehat{D C A}=180-\widehat{C A D}-\widehat{A D C}=180- \widehat{C A B}-\widehat{A D C}=180-20-140=20^{\circ}$. Thus, triangle $C D A$ is isosceles at $D$, so $D B=D A=D C$: triangle $C D B$ is also isosceles at $D$. Since the sum of the angles in triangle $C D B$ is $180^{\circ}$, we have $180^{\circ}=\widehat{D C B}+\widehat{D B C}+\widehat{C B D}=2 \widehat{D B C}+40^{\circ}$, so $\widehat{D B C}=\frac{180-40}{2}=70^{\circ}$. Thus $\widehat{A B C}=70^{\circ}$.

Alternative Solution $n^{\circ} 1$ Since $\widehat{C D B}=40^{\circ}$, we get $\widehat{C D A}=140^{\circ}$, and since the sum of the angles in a triangle is $180^{\circ}$, we get $\widehat{D C A}=20^{\circ}$. Thus, triangle $C D A$ is isosceles at $D$, so $D B= D A=D C$: the midpoint of segment $A B$ is the center of the circumcircle of $A B C$. Thus, $A B C$ is a right triangle at $C$. The angles $\widehat{C A B}$ and $\widehat{A B C}$ are therefore complementary: $\widehat{A B C}=70^{\circ}$.

Grader's Comment: The exercise was generally well done by high school students. Many students confuse the difference between a median and an angle bisector, which are, most of the time, two different lines. Students who chose to use the properties of the central angle rarely completed the exercise, indeed, if $\widehat{B D C}=2 \widehat{B A C}$, this does not necessarily imply that $D$ is the center of the circumcircle of $A B C$. To avoid losing points, students are advised to prioritize elementary angle chasing and to draw figures that were rarely observed in their work.