Triangles can be difficult if introduced without a solid foundation in midsegments and bisectors. To move on to more advanced work, students should understand triangle Angle-Sum Theorem and how to identify medians, altitudes, angle bisectors, and perpendicular bisectors, with an understanding of angle-side relationships in triangles, the exterior Angle Theorem and the Exterior Angle Inequality and work related to Triangle Inequality Theorem. High schoolers by grade 10 should know how to construct the circumcenter or incenter of a triangle, the centroid or orthocenter of a triangle and be thinking about triangles proofs for congruent triangles including isosceles and equilateral triangles. For example, proofs involving corresponding sides/angles in congruent triangles.
It follows that SSS and SAS Theorems and how to prove a triangle is congruent using SSS and SAS are key skills. However, ASA and AAS Theorems should also be studied, and and especially SSS Theorem in the coordinate plane. The Hypotenuse-Leg Theorem should be covered.
Definitely you should know how to identify parts of a circle and the central angles inside a circle with arc measures and arc length and the sector area of a circle. Arc length should be taught with radians and degrees and area of sectors and how to identify chords. A good math syllabus will cover tangent lines, perimeter of polygons with an inscribed circle and inscribed angles, as well as angles formed by chords, secants, and tangents. How to draw a tangent line to a circle and to construct an equilateral or hexagon triangle in a circle is critical knowledge for students planning math-based majors.